What are the best fruits for brewing fruit beers?
Some fruits appear to be classics like cherries raspberries. Some fruits you rarely hear about like kiwi and papya.
Which fruits have flavor profiles strong enough to add to beers and which don't?
There's probably a reason you haven't heard of people using either kiwi or papaya in brewing. Kiwi, papaya, pineapple, melon, and fig all contain enzymes (proteases) that affect proteins. Harold McGee's On Food and Cooking treats this subject. Papaya has long been used as a meat tenderizer, albeit an imperfect one.
The McGee piece addresses mainly these fruits' action on animal proteins, but it's entirely possible that you don't hear about these beers because they fail. Proteins are essential for head retention at the very least, and I'm sure they figure a lot more in the brewing process than I will ever understand.
Many of these fruits were not available in fruit-beer-producing countries at the time that fruit beer styles were being developed, but fig has been available all over Europe in dried form since at least Roman times. It would be a neat taste to add to a beer, and the fact that it hasn't arisen is (shaky) support for the theory that protease-containing fruit are not good in combination with beer.
It seems stone fruit and berries are the most oft-used beer flavorings, probably due to their high flavor-to-volume ratio. I'm sure there are other undiscovered flavoring fruits out there, and I'm not one to stick only to tradition. Mangosteen or rambutan might be awesome beer flavors, but fruit chemistry is at least as complex as brewing chemistry, so there's a decent amount of trial and error or scientific inquiry to be done here.
It might be worthwhile to look at what fruit flavors infuse well in vodka as a pointer to possible other beer flavors.
I know that kiwi-fruit is used in beer. I'm pretty sure that Sam callagione from Dogfish has a kiwi Wit beer in Extreme Brewing. From the little I know about the properties of fruit I think it has a lot to do with the pH of the fruit. Papaya has a higher pH from memory and the better flavouring in beer (citrus, berry, etc) have a lower pH.
My suggestion would be to experiment with small batches to see what works and what doesn't. A 1 gallon extract light ale with whatever you choose to put ihn it isn't going to cost a bomb or waste too much time, but will give you a good idea of what the resulting flavour.
You could even do a full 5 gallon batch and maybe split it 10 ways in to half gallons each with a different fruit.