Dogs respond to the vowel sounds more than the consonants. If you change the name but keep the same (or similar) vowel sounds, chances are it'll be completely seamless.
My uncle got a rescue dog named Sasha. My uncle was mentally handicapped, and had some trouble enunciating, so he started off with this sounding more like "Shasha". He and my mum decided to rename her to "Tasha", which he could pronounce. The dog never noticed the difference.
More of a problem, a number of years earlier, were the names of my gran's dog, my uncle's previous dog, and my sister. My sister is called Jenny, my uncle's dog was called Penny, and my gran's dog was officially called Venus but was always called Venny. My sister of course could hear her own name, so she didn't come when the dogs were called. But if someone called my sister, you'd typically also get both dogs arriving as well!
Or of course you can train them to recognise a completely new name, as Mario says. That'll take a little more effort, but it's not too hard.