Obviously this can vary depending on the version of the story you are referring to, though most of the answers to those variations will be ‘he has no name’.

But the tl,dr here is Henry or Kit.

I assume you are referring to the prince in Disney’s iconic animation of 1950.

Again, he had no official name at the time of the film’s production and release. He was simply billed as ‘the prince’. He wasn't even referred to as Prince Charming until later.

There are rumours that there was a working name for him used by the crew (and possibly cast) in files of story and art from the production, though this is unsubstantiated. That name is said to be Frederick.

Apparently in 1971’s Disney on Parade [1] he was called Prince Alto August Ferdinand. (Or Otto? There’s some confusion.) But this does not appear to have entered the canon at any point, and is for all intents and purposes lost to the ages.

He was something of a minor player in the film to be fair, just as he is in the tales it was based upon (he was originally meant to have more characterisation but it was cut). Same goes for Snow White’s prince, who is even more absent than this guy.

However, as of the introduction of the Princess Collection, these heretofore nameless sods have been retroactively christened. The animated Cinderella’s prince is now officially named Henry. (For those interested, Snow White’s man is Florian.)

In their live-action reboot of 2015 however, the prince is called Kit, though it is never really made clear whether this is a nickname.

Otherwise, you can just give him any name you wish.

Footnotes

Dipende da quale versione della storia stai seguendo. Il Principe Azzurro è abbastanza comune però.

Personalmente, il mio preferito è da Cenerentola di Rodgers & Hammerstein e sono cresciuta con la versione del 1997:

Christopher Rupert Wendimere Vladimir Carl Alexander Francois Reginald Lancelot Herman Gregory James