Red oak is the classic choice in Central California, but just about anything would work well.
If you're using a heavy smoke flavor like mesquite wood, it would be ok over an open grill. Mesquite wood might get a little acrid tasting for long smoking in a closed cooker unless you go really easy on the wood..
Oak Oak Oak Oak Oak Oak !!! I love it. I use some red wine oak barrel strips and it's the best IMO. I've used mesquite before and didn't like it and I'm not a mesquite hater either. I use Royal Oak lump and a 3" x 4" or so strip of the oak.