Wow, I think I had the same thing.
At first I thought it was because I was in the sun too much (I was on a swim team) and maybe it was a cut on my lip that got that way after several days of sun. Then I thought it was a mole of some kind and I was stuck with it on my lips ;_;
I tried rubbing the skin with a washcloth(stupid idea) to see if that would get rid of the skin on my lip with the brown spot, no luck.
For 3 years I made sure I had some sort of lip color or even a colored gloss so it could not be seen.
Last year (when I started college) My mother looked at it and said it looked DARKER! I had stopped paying attention to it as often and ran to check, ACTUALLY I thought it was lighter. Last fall same thing came up, she said it was darker, but it had gotten even lighter. By the spring it was gone! Faded away!
If its not raised or super dark (you may want a doc to look at that) then dont give up hope! Mine took a few years, but it did fade away. In the meantime use lip color to hide it if you can, and I would (and do out of fear that my spot will come back) use lip balm with a good SPF to keep it safe when you are out in the sun.

another good thing to know is that people dont notice it as much as you do. Sometimes someone thought I had chocolate on it (happened like 3 times in 3 years though), but everyone for the most part didnt notice it at all!