Does the UP in 1-UP stand for anything?

There may be extremely vague meaning behind it, according to Wikipedia.

"1-up" was first seen in multi-player pinball and other arcade games. In these games, "1UP" meant that it was player one's turn. Likewise, "2UP" meant it was player two's turn, and so on. In some cases, arcade games also used this terminology to designate which score was whose. "1UP" followed by a score indicated it was player one's score, for example. It is believed in some circles to be short-hand for "player 1, step up to the machine," since a standard pinball table only has enough space and control mechanisms for one player to operate at a time.

From there, it probably caught on as just "1 more life / try".

It's a phrase that originated in pinball machines. There is no hidden meaning as far as I can tell, it's just something someone came up with and caught on, like so many terms.