So why did these White Huns not conquer Persia in reality? The honest answer is we don't know. Information on the Hephthalites is sketchy at best, and so most of this post will be simple speculation. It seems the decision not to conquer Persia early on when they first invaded was a wise one, as it gave them time to establish themselves in India. Had they simply conquered Persia in 485, they likely would have been subsumed by Persia similar to how many of the foreign tribes which invaded and ruled China became Chinese. Looking forward a few decades presents an interesting point of divergence, but before I get that, I should probably briefly lay out the history of this time period.
In 483 the Hephthalites invaded Persia and went on a two year rampage. The Sassanid Emperor at the time was Peroz I, who was killed on the battlefield in 484. His army died with him. In the ensuing power vacuum, his brother Balash seized the throne. Balash paid the White Huns enormous tribute to leave. His four year rule was marred by a power struggled with the sons of Peroz. Zareh's rebellion was quashed with the aid of Armenia, but Kavadh I would prove much more difficult. Kavadh had married a daughter of the Hephthalite King, led a Hephthalite army into Persia with his father-in-law's blessing. With the army outside the gates of the capital, Balash was blinded and deposed by a group of priests and nobles who opened the gates for Emperor Kavadh.
Once established, Kavadh looked for a way to check the power of his magnates, those people who had overthrown his predecessor. He decided to support a religious group known as the Mazdaki sect, which advocated rich men divide their wives and wealth with the poor. The magnates saw through his game and imprisoned him in a tower in Susa. His brother, Djamasp, became emperor and ruled briefly from 496-8. But Kavadh was able to escape the tower and returned in 498 with 30,000 troops from the Hephthalite King. His brother abdicated and Kavadh began his second reign as Emperor of Persia. In that year he had to pay tribute to the Hephthalites, probably in part for their assistance in placing him on the throne. Kavadh could not pay and sought subsidies from Rome, which not so long ago had been subsidized by Persia. Emperor Anastasius refused, hoping the two eastern empires would turn on themselves.
In our timeline, Kavadh managed to forestall the payment of tribute by persuading his father-in-law to support a war against wealthy Rome. But what Anastasius was right and the two eastern empires did turn on themselves? It seems reasonable to think that Kavadh's familial relationship to the Hephthalite king was important in keeping the White Huns from riding into Persia to collect the tribute themselves. But what if Kavadh was not in power? What if when his magnates overthrew him in 496, they chose to execute him?
Executing the son-in-law of Persia's most powerful neighbor, the king to which they paid regular tribute, would not have been a smart idea. The political misstep might have been taken by the Hephthalites as a pretext for invasion in 497. I highly doubt Djamasp, who seemed to have been a puppet of court interests, could have held off the Hephthalite army. By 500 might the White Huns have been in a position to annex Persia? I don't see why not. Now they do not have a puppet to place on the throne, so the Hephthalite king might have seen it as convenient to conquer Persia, organize it as a province in his empire, and perhaps give it to one of his sons to rule.
Organizing such a large empire will be a significant challenge for the Hephthalites. At this time the capital of the Empire was Baktra (Balkh) in modern northern Afghanistan. This is fairly centrally located, though the western end of the Sassanid Empire, where the capital of Ctesiphon is located is quite far away. A system of regional capitals might be adopted. In our timeline, Sakala (Sialkot) in the Indus River Valley became the capital of the Hephthalite Empire. In this timeline in might serve as a regional administrative center for the Indian province. Old Damghan of Parthian fame could serve a similar purpose for Persia.
I am skeptical that this empire would have any sort of longevity. If the king were to give his sons each a province, the empire could quickly disintegrate at his death. Or perhaps after each bloody succession crisis the empire would be reunited. It's really impossible to conclusively decide one way or the other, though it should be noted that large land empires, especially this early in history, never lasted for long. Then in about a century or so, the now sedentary Hephthalites will have to deal with the Göktürks marauding down from the steppes. Perhaps that would signal the end of the empire, with Persians and Indians rebelling to form their own states.
But let us look beyond Central Asia. How does this change effect the rest of the world? The absence of a Roman-Sassanid war in the early sixth century means the Roman (Byzantine, if you prefer) Empire begins the sixth century in a much stronger position. I see no reason to make significant changes until Justinian's reconquest of the west. In this timeline the great general Belisarius would not have had any eastern wars to cut his teeth on, unless of course Anastasius, Justin, or Justinian decided to attack the Hephthalites. But if there was a long period of peace, it is possible Belisarius might not have been the general he was in our timeline. Then again, how can we say that this economically stronger Roman empire would have had a tougher time retaking the west. On the contrary, with more money and no western distractions, I think the Italy and North Africa could not just be retaken but held onto. And perhaps Spain also, which Rome already controlled the coast of, and which had a natural border against the Germanic Kingdoms with the Pyrenees.
Removing the Sassanids also has a significant effect on the balance of power in Arabia and east Africa. The Persians controlled both sides of the Persian gulf directly and had effectively vassalized the tribes in Yemen and Oman. They used those tribes in proxy wars against the Kingdom of Aksum, Rome's Ethiopian allies who had controlled the lucrative Red Sea trade for the preceding three decades. In our timeline, the sixth century saw the Sassanids wrest that trade route from the Aksumites, who were also suffering due to the effects of climate change, overfarming, and state collapse brought about by a number of factors contingent on the loss of control of the Red Sea. If Persia's influence is removed from Arabia then Aksum has a chance to take a breather and, perhaps, regain its footing on the Arabian peninsula. This has huge butterflies, the most notable being the shifting circumstances which may impact the Arabic religious movements which Muhammad was a product of.
I have hardly scratched the surface with this POD, but I think this is enough for now. Let me know in the comments what you think of my ideas when it comes to the effects of White Huns in Persia.