How would you redo the grading of a brick patio?
Best is probably a combination of french drain and pulling it all up
   and regrading the base. Also the most hassle and expense, but in
   terms of function, the best option. And you would then have the option of running the drain under the patio, if that helped with discharging it.

The individual pieces of that that you have thought of are somewhere else along the expense and hassle line, as well as the effectiveness line. Obviously pulling up a few pavers and repacking under them is least expensive, but it will help the water to move away - but if the patio as a whole is not sloped to drain well, it won't help much if you are just moving the puddles around. So, if just a few spots are low, it might be all that's needed, but if filling one low spot will simply result in another, regrading the whole thing looks better eventually.