I was looking for experiences of what was normal a few days ago and couldn't find any so I wanted to leave this behind because DS now has a lovely normal tummy button! 
DS's tummy button has always had a red balloon coming out of it and has oozed green/yellow gunk constantly. I showed it to the GP at 8 weeks and she said to come back at 12 weeks if it hadn't gone and it would be burned off with silver nitrate.
I recently read this NHS link (as well as hearing anecdotes) about babies tummies being burned by mistake when silver nitrate is used and how umbilical granulomas can be treated safely and easily at home with simple salt. Also discussed here on the health visitors website.
I gave it a try - I cleaned the area with cotton wool and water, put a pinch of salt on it (just ordinary table salt from Tesco), cleaned off any excess which was had landed on his tummy, left it for 10 minutes (just distracted him during this time) and then cleaned the salt off as best I could. I did this at approx 7 am and 7 pm. Within about 24 hours, there was a black patch on his lump, and a tiny bit of blood was dried around it, but he was perfectly happy and still couldn't feel it at all. Within 42 hours, it had turned completely black and fallen off (it continued to give out a tiny bit of blood in this time which I would find dried on his vest). Now it is good as new!
Obviously had it been infected (look for red around the area and/or a smell) I wouldn't have touched it myself.
HTH's someone eventually. 