Since the character was a martial arts actor and they needed his name to sound as Hollywood as possible, it's very likely that they borrowed "Cage" from Nicolas. You know, the same name he chose a decade earlier to AVOID reminding people of someone famous.
This is just speculation, but keep in mind that before Nicolas became a celebrity, the name Cage was most commonly associated with an avant garde composer. We seriously doubt the creators of Mortal Kombat were trying to be evocative of a guy playing the piano with a pineapple. Also, according to Wikipedia Johnny Cage's real name is Carlton, which means that, like Nicolas, he was definitely teased as a teenager.
As we've mentioned before, Nicolas Cage is a ginormous nerd. He is such an inconceivably colossal nerd, that he named his son Kal-El after Superman, one of his two favorite comic characters. The other one is Johnny Blaze, the Ghost Rider, of whom Cage even has a tattoo (which had to be covered up when he played Blaze in the Ghost Rider film, because what kind of douchebag has a tattoo of their own face?).