"Hi mir4ea, I'm sorry but I'm still having some difficulty following along. I think you mean you want the script to work in the other way that I mentioned though, that all the selection would get the same unique name assigned to it, is that correct?"

Yes exactly!
For example I have 3 pices of geometry - 1,2,3. I want select them all and push shurtcut to make them one object with unic name :)
Sorry I have to discribe it this way from the start...