To do that, you will have to do one of two things:
1) Rename the existing file to something else first -
File.Move[
^] will do that for you, but you may have to check if that file exists as well.
2) Use a different name.
In either case, check if the file exists with
File.Exists[
^] first.
To use a different name, I'd suggest that you append a version number just before the file extension - and if you use an integer value to can check if it exists, and increment it if it does, then recheck.
So you would end up with
MyPic.jpg
MyPic.0001.jpg
MyPic.0002.jpg
...