Finally finished removing the grid lines around my friend's world map for his Christmas present!

1) original image
2) original image with the lines removed
3) filter applied to remove some of the gross areas that were much harder to get rid of
4) color reintroduced, final image

I think it looks pretty decent given what I had to work with and it kind of gives it a more "drawn map that has worn over time" vibe than the original which was more clearly just made with software.