What are Log Stacks

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.

Option 1 is definitely not the definition - in the scheme setting, it doesn't yield log structures on the divisors in question, and you don't get log points. A.J. mentioned that you probably meant to say, "stack equipped with a log structure". You can find a definition in section 5 of Martin Olsson's paper Logarithmic geometry and algebraic stacks. His web page doesn't have it, but you can find it on Google Scholar.

Option 2 works also. Fine log schemes over a base fine log scheme form an algebraic stack over the underlying base scheme. You can do your log stack theory over this. See Corollary 5.8 in Olsson's paper comparing the two options.