Well, I don't know exactly why it was never mentoned in the film,but I suppose it was becuase the film was supposed to be focusing on Snow White and the seven dwarves and Walt Disney didn't think it was important to give him a name. However, Disney has given him a name (like they gave Princess Aurora's mother the name Leah, and Cinderella's prince is called Prince Charming) and his name is Prince Ferdinand.
At the 1938 Oscars, Shirley Temple mentions the name "Ferdinand" among others while presenting Walt Disney his awards. Many have interpreted that she was referring to the Prince, saying that Ferdinand is his name. However, she was not referring to the Prince, but rather to Ferdinand the Bull. As confirmed by Dave Smith, Historian of the Walt Disney Studio Archives, the Prince was never given a name in the film and "Ferdinand" has never been used as one by the company. Despite this, many Disney fans still believe that the Prince's name is Ferdinand. 
Disney Latino revealed through a video of Top 10 Romantic Moments the name of the prince is "Fernando" Spanish translation of the name Fernidand.
Disney France once revealed in a TV-spot that his real name is Florian. That same TV-spot also revealed that Prince Charming's real name is Henry (or "Henri," as it would be in French).
The name "Florian" originates from this article. It was allegedly used by Disney when it started the Disney Princess franchise.
There's a story that there are artwork and documents in the Walt Disney Studio Archives that call the Prince "Frederick".