The more real wood you put in there the better. I am no guitar tech, but been working on that, and I've worked with wood a lot in general. Especially in a high stress point - like a strap button - I'd properly fill that with wood. A glue bond of wood to wood is actually stronger than wood, but a clump of glue in a hole (even with sawdust) on it's own will have no strength.

Also, I would use a nice long screw after the repair. Not sure what was in your's, but the screws for buttons I purchased for a build were only 1". I think I exchanged them with 1 1/2" (you'll have to drill the hole to accomodate).