1st, you cannot replace the cart connector on the mini, so scratch that off your list of to-dos.
2nd, like he ^^^ said, buy some Gamebits. They're an entirely worthwhile investment.
3rd, what are the symptoms? Have you tested the AC adapter to make sure that's not the issue? Does the TV flicker when you power up the SNES?
Here's what I use to clean cart connector pins. Works about 95% of the time.
http://www.crescentartists.com/artbo...tboard_ill.htm
Illustration board comes in quite a few thicknesses and can be cut down to little pieces that are the exact size of the cart connector that you're trying to clean. The usable surface side of the cold press board has a good amount of texture in it so it's quite effective for pulling crud and debris from the pins. You can even add a little isopropyl or other cleaner to the board if you like. Use it like any other cleaning kit, which is done by inserting the board into the port over and over again.
If that doesn't work, then the last ditch effort is to wrap a piece of 1500-2000 grit sandpaper around the board and give it 3-4 insertions into the cart port and then try a game. If still not working, then repeat. If still not working after 2-3 rounds, then you very likely have a different problem.
The only place you'll find a replacement SNES mini button is to scavenge it from another, unless you can find someone parting out SNES minis on eBay or the like.