If you delete the record first, the name is gone. If you show the message then the delete fails for any reason, the book is still OnLoan even though it has been checked in. Consider flagging the record instead of deleting. Not sure if you have another means of tracking loans, but this would also allow you to report on patron loan frequency, loan counts on any given book, or anything else you can think of.
EDIT
A query or DLookup can do what you want. You've posted in the Forms part of the form, so I imagine you have one to work from as was assumed. However, if you have more than one copy of a book, how are you tying the returned copy to Mr. Smith instead of anyone else who has borrowed one?
The more we hear silence, the more we begin to think about our value in this universe.
Paraphrase of Professor Brian Cox.