This is a very interesting scenario. One important question is when exactly do the Hephthalites conquer Persia. Before speculating, I want to lay out the historical scenario (as I understand it, some of this could be wrong).
In 483 the Hephthalites invaded Persia and went on a two year rampage. The Sassanid Emperor when the White Huns invaded was Peroz I, who was killed on the battle field in 484, his army wiped out. His brother and successor, Balash, paid the White Huns an enormous tribute to leave. He ruled for four years before Kavadh I, a son of Peroz, rose in revolt. Kavadh apparently had married a daughter of the Hephthalite King, and acted with his father-in-law's blessing at the head of a Hephthalite army. With the army outside the gates of the capital, Balash was blinded and deposed by a group of priests and nobles and Kavadh installed as emperor.
Kavadh supported a religious group known as the Mazdaki sect, who advocated rich men divide their wives and wealth with the poor. This obvious attempt to curtail the magnates who had overthrown his predecessor failed and Kavadh was deposed by the magnates and locked away in a tower in Susa. His brother, Djamasp, became emperorIt should be noted that 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 a tribute to the Hephthalites, perhaps for their assistance in placing him back on the throne, perhaps it was just an annual demand. But regardless, Kavadh could not pay and so sought subsidies from Rome. Emperor Anastasius refused,in hopes that the two eastern empire would turn on themselves.
I should note that the information I have been able to turn up is all very sketchy on the details. So a lot of these connections are tenuous, but it seems plausible to me that though Kavadh could not pay tribute, he managed to prevent conflict with the Hephthalites (who, remember, were ruled by the man who was his father-in-law) by suggesting a war against the rich Roman Empire. The Hephthalites agreed and in 502 hostilities began around Armenia. The war was interested by an invasion of Western Huns (not Hephthalites) through the Caucuses, necessitating a peace treaty wherein Rome paid the Sassanids tribute. Presumably most of that tribute went straight into the coffers of the Hephthalites.
From there on the relationship between the Sassanids and Hephthalites becomes much more difficult to ascertain. I'm going to stop my historical summary for now, because I just had a nifty little POD idea. Suppose that Kavadh was murdered, rather than imprisoned. Then in 498 when the Hephthalites come demanding tribute, Emperor Djamasp has no familial connection to prevent an invasion when it becomes obvious they cannot pay. Rather than a Roman-Persian War in 502, we have a Sassanid-Hephthalite War in 499, a quick and total victory for the Hephthalites, who, this time, decide to set up shop in the Iranian plateau permanently instead of just plundering and leaving.
This is massive, to say the least. The Hephthalite Empire is now one of the largest land empires in history up to this point. Of course, it is no sure thing that the Hephthalites could hang on to establish a lasting dynasty, but I don't see why not. Unfortunately I don't know enough about the White Huns to speculate on what their empire might look like or do. Any ideas?