What is the easiest way to replace an existing backsplash?
Look on the right side of your picture.  You can tell the granite and fake stucco finish is layered on the drywall.  

First you will not get drywall or backboard to adhere permanently to granite so it has to go or you have to use it.

Personally I would keep the granite.  Then I would scrape off all of the stucco.  It might sound like a pain in the ass but its not a HUGE project - 3 or 4 hours.  Then you might have to do some minor repairs to it.  If anything I would just take this stuff off in chunks and skimcoat the area afterwards.  

You don't need backerboard unless you want to put up a heavier stone.  Drywall is fine.  Just scratch it up good before hand.  Throw down some tile that fits in with the granite.  I would use thinset instead of mastic so that I can level out any wall issues as I go.  

If you keep the stucco and put something over it you are going to have some really odd edges and it just won't look right.