I think it depends on the countertop and personal taste-- shinier or matte. If you pick honed, I imagine you'd have to seal it just like a countertop.

What does the rest of your kitchen look like?

The kitchen will have Simply White shaker cabs and Calico White MSI quartz counters. I'll have a stainless undercount sink, chrome faucet and drawer pulls. It is a small kitchen about 10x10 feet.

I'd go with honed because I wouldn't want the reflectivity of the polished on the vertical surface.

Definitely honed. I put in polished backsplash which is too hard and shiny looking for a home.

I think you already have nice contrast with shiny vs matte with the chrome pulls-- my sense is that honed would look better with your quartz countertop. I'm going to do the same thing with a marble hex inset above the range and I prefer the honed finish. You may want to see a sample of each with the countertop to be sure.