a song of ice and fire - Does one need to be Targaryen to hatch a dragon's egg?

I've looked, but I haven't found any evidence indicating that one needs to be of the Targaryen name (or even of the Targaryen blood) to hatch a dragon.

You need fire and (a sacrifice of) blood, as the House words describe, but that's where the relation to House Targaryen appears to end.

That being said, the whole immunity to fire thing probably makes raising said dragon much, much easier.

Answer

No one doesn't need to be a Targaryen.

In ancient Valyria, Targaryens were one of forty Freeholder families who all owned dragons. Targaryens were one of the most minor Valyrian families. Before Doom of Valyria, they fled to Dragonstone and escaped the fate that befell the rest of Valyria, with her proudest dynasties. So since all of them had dragons, it should be clear that being a Targaryen is not needed.

In Westeros, Princess Rhaenyra's sons Lucerys, Jacaerys and Joffrey successfully hatched dragon eggs even though all of them were Velaryons¹. It was rumored that Dragons wouldn't hatch but they did.

It must be noted however Velaryons were also Valyrians and the Velaryon princelings were half Targaryens. So from canon evidence, only being of Valyrian blood is needed. But all Valyrian sorcery was rooted in fire and blood so technically anyone familiar with blood magic could hatch a dragon.

The Targaryens are not immune to fire, many of them like King Aegon V, Princess Rhaenyra, Prince Duncan the small, Prince Viserys, Prince Aerion brightflame etc died due to burning. The Dragonlore is not exclusive to Targaryens either.

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^{1. There were also rumors that the Velaryon princes were not actually trueborn sons of Rhaenyra Targaryen and Laenor Velaryon. They were actually bastards of Ser Harwin Strong. The evidence was in outlook of the children, while both parents were Valyrian, none of the children had traditional silver hair, aquiline nose, blue-ish eyes, instead all of them had common Strong features.}