Funny you should ask....
I just took the kids I used to babysit mini-golfing today, after months of begging from the littlest guy. There are two girls (12 and 10) and one boy (he's 5). I don't have any feedback regarding how early to start them at mini-golf (except that they ALL loved it!!), but I thought I should warn you...
Be prepared to lose if you're not a mini-golfer, because the lowest score in our group (and, therefore, the WINNER) was also the youngest!!
Yes, a 5 year old beat me at mini-golf, completly legit - there was no sympathy, there was no cheating. He beat us all fair and square, and he'd only been mini-golfing once before.
There's SO going to be a rematch.