How to solder this broken Gameboy Color part back on?
If the solder did not stick, then your soldering technique was likely not correct. First, I would remove the old solder. Use a solder sucker or copper wick to remove the old solder. Clean the pad with isopropyl alcohol or contact cleaner. You could also add a bit of flux if you still have issues with getting solder to stick.

Once the pad is clean, ensure your soldering iron tip is clean and tinned. Heat the pad for a couple seconds, and apply solder to the pad (Not the tip of the iron!). Place the GBC part in place and heat up joint until the solder melts, keeping heat on the joint for another second or two. That should be it.