So, there's this dude named Pericles, and we follow his adventures throughout the play. Also, he's the ruler of Tyre, which is a pretty big deal. Technically, Pericles is a king, but characters in this play toss around the terms "prince" and "king" like they're interchangeable. (Both just mean that he's a monarch.)
At times, though, it seems like Pericles is referred to as a "prince" when Shakespeare wants us to know that the guy is still pretty young and has a lot of growing up to do—especially at the beginning of the play, when King Antiochus is all, "Young prince of Tyre, you have at large received / The danger of the task you undertake" (1.1.1-2).
Translation: "Hey, kid. Good luck trying to guess that riddle. I'd hate to have to lop off your head and mount it on my wall if you guess incorrectly." The interesting thing here is that Pericles really does have a lot to learn at this point—he has absolutely no idea what he's gotten himself into at Antiochus's court.
The "prince" part of Pericles's name, then, draws attention to the journey our hero has to undertake before he can actually get what he wants: a wife, a family, and a happy kingdom.