This name is also a nickname. My grandparents call me this all the time. They speak very fluent Cajun French and in it, 'Cher' (along in regular French)means 'love' or 'dear'. This name I've rarely seen given to girls though, and, like I said, the only time I've ever heard it was to me by my grandparents.
This name will probably be associated with the singer for a significant amount of time. I wouldn't recommend it on a real child, especially as a legal name. I don't find anything wrong with Cher, but when a famous person is known by only one name, that name becomes associated with them.
My name is Cher. I wasn't however named after the singer, I was named after my mum who is called Cheryl. I love my name and I love how unique it is. I have had no problems with it and I love it very much :)