This question was answered in Martin Olsson's thesis ( http://math.berkeley.edu/~molsson/thesis.ps ). He gives sufficient conditions for a fibered category on the category of log schemes to arise from your option 1, which he calls a stack with a log structure.

This question was answered in Martin Olsson's thesis ( http://math.berkeley.edu/~molsson/thesis.ps ). He gives sufficient conditions for a fibered category on the category of log schemes to arise from "a stack with a log structure", which I think is more or less what Charles intended by Option 1.