I pretty much agree with everyone else about if she pays for the day then it is hers. That being said, if this has been a long term arrangement that she NEVER uses these days then why wouldn't you use that to your advantage and make appointments. I think moving forward there is a pretty easy solution to this and it really is just about good communication.
You are not a drop in service so if she is going to attend the daycare on a Friday that she isn't scheduled to come, paid for or not, there needs to be a minimum amount of notice she gives you to say she will be coming. For example, if she never uses those days and the child is your only one who pays for that day, you would have no activities prepped, no food prepped so I would request 24 hours notice if she intends to attend so you can be prepared for the child. It's not that he can't use the day if she were to tell you at pick up the day before, after all she pays for the day, but without notice you are not prepared for him and if she is only going to sporadically use the days how does she expect you to plan?? This would show the parent that you like to be organized for each day to be able to meet the child's needs without last minute rushing or unnecessary preparation if they aren't to attend that day, and it also gives you 24 hours notice to cancel any appointments you may have made without a financial penalty.....does that make sense or did I ramble?
Yes it is her day as she pays for it, but are you expected to get up in time for opening time for just one kid on the rare off chance they will show up? That is ridiculous. She either needs to show up every week or show you the courtesy of some prior notice so you can plan the day in advance, whether that be food prep, activity prep, or cancelling scheduled appointments. I don't believe you can deny her the day if she paid for it, but you can set some new rules in your contract that work in both parties favour.