First of all, we would say "that's a beautiful name", unless, say, you were trying to pick a baby name and while looking down a long list of possible names you find a nice one and, pointing to it, exclaim "this is a beautiful name!"
Secondly, I know that while in English, "name" by default refers to someone's personal name and you have to specify if you mean their family name ("last name", "surname", etc.), in French it's the other way around: nom means "name" but is assumed to mean "last name" unless you specify prénom. Does Hungarian work more or less like French in this respect?
I don't think so. "Név" in itself, usually means the personal name. Or the full name.
Btw, in Hungarian, the family name comes first, then the personal name.
The family name can be called "családnév" or "vezetéknév". The personal name is "keresztnév" or, officially, "utónév".