When the backsplash was installed, the hole left for an outlet box is a tad too small. What is my best option to enlarge the hole to slide the box in?
An angle grinder with a diamond-dust blade will cut that tile easily. It's very messy, of course, so have a helper hold a strong vacuum right near the blade as you cut.
You might also be able to just file enough material away to get it to fit. A common rasp or other abrasive tool could work. Wall tile is relatively soft, but the glaze isn't, so expect some dulling of such tools to occur. Don't use your best woodworking equipment.