I know it's a stretch, and I'm pretty sure it's not, but is 名前 related to name? I always thought it was weird that they were so similar in pronunciation.
It's just a coincidence and an example of a false cognate.
The etymology is covered here in Japanese.
Basically, the term "名" has been around for a pretty long time with the same meaning as 名前. It's thought that the 前 part is an honorific that was added some time later. Early uses of the full word 名前 can be seen in use in relatively modern times. The English term for it seems to be "modern" in the historical sense, but in Japanese 近世 seems to refer roughly to the Azuchi-momoyama period through Edo, roughly 1500s to mid 1800s. Not quite sure exactly when the first recorded use was. It was widely used from the Meiji period onward.