As it stands, one of the biggest mysteries in the story of Bleach is the nature of the Hogyoku. Even though Hogyoku were created by Urahara and Aizen, we still do not know what the Hogyoku is or how it was made. What we know of the Hogyoku is that it is a substance "that reacts beyond the conventional physics of Soul Society" (Aizen Sousuke, chapter 175), that it possesses a will of its own and its true power is "to manifest into reality the heart's desire" (Chapter 401). In Aizen's description of the Hogyoku's power in chapter 401, he makes an interesting assertion that could explain a number of things behind the events it influenced: Aizen stated that the events "materialized" by the Hogyoku should be called "miracles".
By definition, "miracles" are either events not explicable by natural or scientific laws, or are events that are statistically unlikely yet not contrary to the laws of nature.
While he did state the Hogyoku is more like "a power that guides in the direction of desire", because the Hogyoku cannot manifest the heart's desire if the target essentially lacks the power necessary to beget their desire, it does not take away from his description of the events affecting those around the Hogyoku as miracles.
To start off, let us look at the Visoreds. We know from "Everything But the Rain" that Hollowfication generally results in two things: the Hollowfication turning the Hollow-Shinigami hybrid into a monster without reason, and "Soul Suicide" from the destabilization of the soul and the shattering of the boundary between the soul and the external world. The former was a process that could not be controlled, as it was beyond the technologies that Soul Society possessed when they first undertook the experiments. However, the Hogyoku could not only stabilize and make complete the Hollowfication of the Visoreds, who were already undergoing a virulent form of Hollowfication, but it was also able to make Tousen into a perfect Hollow-Shinigami hybrid capable of using Resurreccion.
In other words, Soul Society's technologies were incapable of perfecting and stabilizing the process of Hollowfication, but the Hogyoku's power was able to both perfect and stabilize the process, even if Hollowfication is already occurring in the target. This was one of the first and most prominent of the Hogyoku's "miracles".
What the Hogyoku could not do, however, was prevent "soul suicide", because the targets in both cases inherently lacked what was necessary to do so - Quincy arrows of light and Human souls to balance out the Shinigami and Hollow side. Urahara had to create the vaccine himself and apply it to the Visoreds to restore balance to their souls.
The second situation Aizen cites is Rukia's giving all her power to Ichigo. Ordinarily, it seems a Shinigami should only be able to give half of their Shinigami power to a Human, but the Hogyoku caused all of Rukia's Shinigami power to be transferred to Ichigo instead of the half she originally intended because she was still carrying the guilt of killing Kaien. More importantly, Rukia said the process was supposed to be "temporary", but Ichigo retained the Shinigami power that Rukia gave him long after that night.
In other words, a process that was only meant to give Ichigo half of Rukia's power of the Shinigami temporarily ended up giving him all of Rukia's power of the Shingiami permanently. This is another "miracle" brought about by the Hogyoku's power.
The third event Aizen described was the awakening of Inoue and Sado's powers. There is much debate over this, as the "Lost Agent Arc" formally introduced the power of Fullbring, and people jumped to separate conclusions for both characters. In Sado's case, he's the only one out of the two considered to be a Fullbringer "because he was the only one who was stated as being one", even though visual evidence, and what we know of Fullbring as a whole, also indicates Inoue is a Fullbringer. While many people prefer to think that we do not know "what" Inoue's power actually is, they are perfectly okay with assuming that Sado's mother was attacked by a Hollow before he was born to explain his power being a Fullbring.
After all, we were told in that same arc that Fullbringers are people whose mothers were attacked by Hollows before they were born, resulting in the remnants of Hollow reiryoku in the mothers being passed onto their children, who developed a power derived from Hollows - Fullbring. We now know this to be possible because Humans are opposite of Hollow, as stated by Urahara in "Everything But the Rain", and the two balance each other out.
However, we cannot forget that Aizen grouped Inoue and Sado's awakening of their unique abilities together with all the other events/"miracles" materialized by the Hogyoku's power, which implies that their awakening unique abilities would not have been an event that occurred naturally without the Hogyoku's involvement.
To explain this, let us look at what we know of both their powers, how they awakened, and how they both relate to the Hogyoku:
1) Urahara stated that Inoue and Sado awakened powers because of Ichigo's reiatsu. At best, this is likely a half-truth, because we know that Keigo developed spiritual awareness after being in close proximity to Ichigo and Uryu's fight with the Menos (where Rukia also was with the Hogyoku) but he has not developed any unique abilities like Inoue and Sado. The same goes for Tatsuki who, although she was attacked by Acidwire and began developing spiritual awareness like Inoue, has not awakened any special abilities.
Urahara's statement likely ties into the effects of powerful reiatsu on Human souls. First off, we know that all Vice-Captain and Captain-level Shinigami have 80% of their power sealed away when they travel to the Human World. This is done in order to avoid having their reiatsu unduly affect the souls of Humans in the living world. As a result, this is likely done because exposure to reiatsu at the level of a Vice-Captain or Captain i.e. around 5x greater than Ichigo's reiatsu at the beginning of the series at minimum, would cause a Human soul to develop spiritual awareness.
In other words, while Ichigo's reiatsu was responsible for them developing spiritual awareness, it was not responsible for them manifesting their unique abilities. After all, how would have Sado developed a power closer to that of Hollow's if the source of his powers awakening was a Shinigami's reiatsu?
This brings us to...
2) Both Inoue and Sado were attacked and severely injured by Hollows. We know that there were heavy traces of reiatsu in Sado's wounds when he was initially attacked by Shrieker, meaning that traces of Shrieker's reiatsu would have remained in Sado's body after those wounds healed. We also know that not only did Inoue have a Hollow bite mark on her leg, but she was also bitten by Acidwire when her soul was outside of her body. That's more than enough exposure to a Hollow to result in remnants of the power remaining in her physical body (and soul). As we know from the "Lost Agent arc", remnants of Hollow reiryoku in a Human body is what begets Fullbring.
"But Humans only have Fullbring if their mothers were attacked by Hollows", some will inevitably say, but I do not think that rule would be written in stone with a power like the Hogyoku involved.
I believe it would be a situation like the Arrancar. While there were "natural Arrancar" in Aizen's army, we know that most of the "natural Arrancar" in the Espada were replaced by Arrancar created via the Hogyoku, or Arrancar whose power was "perfected" by the Hogyoku. The Hogyoku-created Arrancar were "perfect Arrancar", and their power eclipsed that of the Arrancar that came before them - the ones that were referred to as "mock-Arrancar" by various characters.
The situation with Inoue and Sado can arguably be said to be the same: while we know the means by which "natural Fullbringers" are born, Inoue and Sado are Fullbringers whose power was brought about by the Hogyoku. We can even say this is why their respective powers have traits the other Fullbringers do not - Sado's Fullbring being the "skin" of his body rather than an item like his Abuelo's coin, or Inoue's Rikka manifesting as fairies able to communicate with her.
In other words, Inoue and Sado's powers could be said to be Fullbring that were "perfected" by the Hogyoku.
Up and coming is the third point...
3) Both of them have had their powers compared to those related to Hollows. We know that Sado came to the conclusion that his powers were more like those of a Hollow in his fight with Gantenbainne, as his power became unsettled in Hueco Mundo and only settled down as a result of the fight with Gantenbainne.
In Inoue's case, it was not a direct comparison to a Hollow's power but a Visored's Kidou (which in part has led to all the debate surrounding "what" her power is). We know that Hachigen developed his unique Kidou after he attained his Hollow power, meaning that Kidou incorporated Hollow reiryoku into its structure and creation. I believe it is this common Hollow element that is why Hachigen said Inoue's Shun Shun Rikka are similar to his own Kidou - both utilize barrier techniques derived from Hollow reiryoku.
As both Inoue and Sado were attacked by Hollows, it makes sense that the Hogyoku would have used the remnants of Hollow reiryoku in them as a base for them to develop their unique abilities.
To bring this section of the post to a conclusion, I believe that the Hogyoku not only allowed for Inoue, Sado, Tatuski and Keigo to develop spiritual awareness as a result of Ichigo's reiatsu, but also for Inoue and Sado to manifest Fullbring abilities as a result of the Hollow reiryoku left over from when they were attacked. Normally, neither event would have happened - it requires exposure to Captain-level reiatsu for a Human to develop spiritual awareness, and Fullbringers are normally born to mothers who survived Hollow attacks in the past - but the Hogyoku's involvement not only allowed them to awaken spiritual awareness and manifest Fullbring, but the Hogyoku could also have been responsible for their Fullbring being "perfected" in comparison to normal Fullbringers.
That is the third "miracle" created by the Hogyoku, as cited by Aizen.
The reason why I believe the circumstances under which Inoue and Sado manifested Fullbring are possible is because of Aizen's Transcendence.
We know for a fact that Aizen, while immensely powerful, was still a normal Shinigami. By the time of Fake Karakura Town, he had fused his soul with that of the Hogyoku's, and he was in the process of helping along the development of Ichigo's power until he reached a certain point - the point where Ichigo's attained a power "even greater than Hollowfication". As Aizen was explaining this, recall what he was doing - he was examining Ichigo's reiatsu which was left behind in a wound caused by a Masked Ichigo's Getsuga.
Ichigo had commented on his new Mask feeling different when compared to his normal Mask, on top of the pattern of the Hollow Mask being drastically different from what it was before. That was the point where Ichigo's power had evolved into "the power of a Transcendent Being". Aizen sought to attain that power, and Ichigo was the perfect subject to use in that effort because of his heritage and circumstances of birth - the child of a Hollowfied Quincy (Masaki) and a Human-Shinigami (Isshin). In other words, Ichigo was born with a perfect balance of all four soul types comprising his soul, an entirely different entity from the Visoreds who required a Vaccine to balance out their Shinigami and Hollow sides.
In other words, the Getsuga wound now caused traces of Ichigo's Transcendent reiatsu to reside in Aizen's body. Once Aizen was pushed to his limits as a Shinigami, the Hogyoku was able to use the traces of Ichigo's reiatsu in Aizen's body to bring about Aizen's evolution into a Transcendent Being. It is not that different from the concept of Hollowfication - pouring a Hollow soul into a Shinigami soul to sublimate the soul to a higher order - or the birth of a Fullbringer - Hollow reiryoku mixing with a Human soul - but it is on a different level entirely.
With little doubt, no process or technology possessed by the Soul Society (or any faction for that matter) would have been able to replicate those results. In other words, the Hogyoku's "desire materialization" was the only way for Aizen to be able to use Ichigo's reiatsu to attain the power of a Transcendent Being, just as the Hogyoku used the Hollow reiryoku leftover in Inoue and Sado's bodies as a base for them to develop Fullbring.
In other words, Aizen being able to become a Transcendent Being (and the immortality granted to him via fusing with the Hogyoku) is another of the "miracles" brought about by the Hogyoku.
In conclusion, the Hogyoku's power allows it to materialize events that would otherwise be impossible, so long as the "potential" exists for it to occur. The stabilization of Hollowfication was originally impossible for Soul Society, but the Hogyoku was able to stabilize both the Visoreds and Tousen's Hollowfication (even if it couldn't prevent "Soul Suicide"); Rukia had only intended to give Ichigo half her Shinigami powers temporarily, but the Hogyoku caused all of Rukia's Shinigami power to be transferred to Ichigo permanently (at least until Byakuya took that power away); while Fullbringers are normally born to mothers who were attacked by Hollows in the past, the Hogyoku allowed Inoue and Sado to manifest Fullbring by using the Hollow reiryoku in their bodies as a base, just as Aizen used Ichigo's reiatsu as a base for his Transcendence.
Because all of these occurrence were either events that were normally impossible, all of these can be defined as "miracles" brought about by the Hogyoku.