C.C.
I personally went through it. My littlest one was only 5 months old when I had surgery to correct my herniated umbilical. I wouldn't say too much to your daughter in advance. Just tell her that the doctor needs to fix her belly button and they will do it while she is sleeping. And explain she will be sleepy for the rest of that day and that her belly button is going to have a boo-boo but that it will feel better in a few days.
My experience is that kids are far more resilient than we adults are! My 9 year old got her tonsils/adenoids removed about a year ago, and yes she woke up hurting and was not happy at all, but after the first day she was quite the trooper!
Good luck!