So in my theoretical example, a group of 50 of so people takes 100kBTC and goes to live on a desert island. They have a local network, but no connection to the internet as a whole. They attempt to run an economy.
What problems might arise? Would mining still work? Could transactions be verified?
Taking today's blockchain with them, and its difficulty of about today's 2.7 million, and assuming each of the 50 in the group has a typical high end 1.4 GHash rig, then to get a block on this isolated network to be solved, you would have to wait two days. So to get a transaction to confirm would take nearly two weeks.
Now consider that someone takes a copy of their local wallet with 500BTC that they earned growing carrots back into the world and tries to spend it. Does it work?
Well, ... all the blocks solved by your island would be orphaned once exposed to the main net because the current capacity of 20 Thashs/s has way more blocks solved than the island fork. But the client wouldn't care, it would see the reorg and notice the transactions were back to being unconfirmed and would broadcast them again. Presuming no double spending occurred then the transactions would be relayed and then included in blocks. So rejoining is not a problem, as long as there were no dishonest who spent on the island and also spent those same coins on the main net as well.
Is there any way to adapt Bitcoin to solve any problems with this scenario?
Yes -- set up something like a dial up connection if you had to for just one of the nodes even so that you are never out of sync very far with the rest of the network.