Show them the lineage they are now a part of. Match the book to the person.
A well-chosen Jewish biography or history book hands someone their inheritance in print. A book about Jewish comedians, athletes, scientists, artists, or revolutionaries shows a young person the vast, wild, accomplished lineage they have just joined, and does it through a story they will actually want to read.
The trick is matching the book to the reader. Find the one that connects to what they already love, and the history goes down easy. A kid who loves basketball, a teenager into science, a friend obsessed with old Hollywood, there is a Jewish story waiting for every one of them.
For a bar or bat mitzvah especially, it is a gift that treats a young person as a thinking adult, which is exactly the message the occasion is meant to send. It says: you are part of something big now, and here is proof.
A bar or bat mitzvah with a specific passion, or any curious reader. Best when you know the person well enough to match the subject to their interests.
Any good bookstore, your local library sale, or the Jewish-interest shelf at a Judaica shop. Used copies are easy to find.