Oak has grain which gives it movement. A backsplash with movement can look too busy. The plain countertop might make it possible to use something with some movement - but don't overdo it - not a good place for a backsplash of tiny glass tiles.

I have oak cabinets, but my quartz also has movement so I kept my backsplash plain with cream subway tile. You could consider a color to add something, or keep it simple and dress up the kitchen with your decor.

Cream subway tile is also inexpensive.

This is mine:

My pics -work in progress: · More Info

forgot to mention that I'm hoping to incorporate more grey/white in this house. The ceiling is exposed wood with beautiful exposed beams...so lots of wood. Grey okay? And would grey subway tile work then?

It can be difficult to get the cool grays to work with warm wood tones - which is why they are often used with white cabinets. However, I have seen it done. You may want to tie the color in with your flooring since that is gray.

This gray tile works here - but it is also reflecting the gray in the counter:

A stone look gray may also work:

Is the countertop in great shape? My concern is that the new backsplash might leave the countertop looking old - and it is hard to replace the countertop after the backsplash has gone in. You might want to get an estimate on a new laminate countertop with some grays and beiges (like my countertop and the first picture above) to just see if it is in your budget. You could also cover the damaged areas of the drywall with Zinsser Gardz (repair deep holes with spackle or drywall mud) and then just paint the backsplash (try gray) and call it a day. That could get you by till you have money for countertops too.

If you want a traditional and classy look to achieve, go with stone look backsplash. Since you appliances are going to be black, you may find Vienna Blend Brick interesting.