"What is your name" is the main clause. "Now that you've changed it" is the dependent clause. However, it is not a restrictive clause (the fact that it has been changed does not affect the understanding of the main clause). Because it is not a restrictive clause, you need a comma after the main clause. "What is your name, now that you've changed it. You can also take the second clause as an explanatory clause, the reason why the question is being asked, and explanatory clauses also get commas before them (admittedly, this last point is probably unorthodox, but I think it still holds true).
Either way, while the sentence reads fine without it, the "proper" method is to use a comma—well, proper according to the Chicago Manual of Style. There may be other style manuals that say different.