I'd lean toward filling with wood. It would be best if you can fill with plugs, rather than dowels, to avoid endgrain facing the weather/sun. If the wood in the holes is blackened/deeply discolored, you might consider drilling them out first (it'd probably be easier anyway.)

Filling with epoxy will work, but is not preferred because the epoxy does not expand/contract with moisture/heat changes at the same rate, and may result in slowly crushing/tearing surrounding wood fibers. (But this is a somewhat extreme wood purist viewpoint. For the same reason, try to use wood plugs of similar wood rather than something markedly different.)