Hi! It could be the power adapter itself that is a problem, or it could be the DC-in port that the plug on the power adapter connects to. Since replacing the DC-in on a 12" PowerBook is a difficult procedure, I'd say the first step is to get a hold of a known-good power adapter and determine whether it solves the problem (which means your power adapter is bad), or the problem still exists (which means your DC-in is bad). If it's your DC-in, let me know, and I can point you to the part and the iFixit guide to replacement.
I would also try plugging the AC adapter into a completely different electrical outlet, preferably on the other side of the house. Sometimes if you have bad wiring, or the electrical port is overloaded, etc., if can cause bizarre behavior, although in your case it's probably not that...but it can't hurt to rule it out.
Also, you mention a "blue light", whereas it should actually be green or yellow. Is it really blue? If so, do you have an actual Apple AC adapter or 3rd party? If it's 3rd party, quirks related to that might be all the more reason to try a genuine Apple AC adapter.
On a side note, I'm curious if the computer works on battery power?