…you don’t want to give cash…and she’s the big boss’ daughter? (the girl seems very sweet…but I’ve met her only once!)
Maybe this will give you some ideas. https://amenvamen.com/best-bat-mitzvah-gift-ideas/
I’ve had invitations custom framed (double glass) and people love the keepsake.
Cash gift is fine. The gift should be in a multiple of 18
Cash is fine. I usually give gift cards to either Amazon or Barnes and Noble.
Cash or check in a multiple of 18.
A bracelet or necklace. There are some at Jewish gift stores appropriate for 13 year old girls. Or bangles like Alex and Ani.
A music box.
Amazon gift cards are what we get kids we don’t know well. Even kids we do.
Just want to chime in that cash or a check in multiples of $18 is very accepted/expected. My sons rcvd mostly checks. The 18th letter of the Hebrew alphabet is chai - which also means life - like the toast l’chaim. So that is why the custom of giving a multiple of $18 for the check amount.
You can also give $100 plus a multiple of $18 – like, $118.
Whatever you do, please don’t give her artwork. We’ve got several prints in the house that were gifts and most of them are not to our taste
Cash is very typical, but if you don’t want to do that, a nice gift box from a trendy teen store like Lush works well.
I have been on a path of an appropriate sterling charm or charms on a slim sterling silver necklace and an Amazon gift card for youngish women occasions. Etsy has some pretty ones if you search on best mitzvah charm for around $50.
Back in the day lots of people gave Israel bonds - they now sell eMITZVAH bonds.
We used to give Jewish history books and such, trying to tie into the religious significance of the occasion. But that sort of fell by the wayside. We gave my nephew an Amazon gift card and he spent it on Monty Python videos - very disappointing that there would be no lasting significant object from us. We later gave my niece a bracelet from a museum gift shop which she apparently loved.
Nowadays I would probably look for Jewish jewelry - a nice Star of David or Tree of Life necklace maybe. Something that maybe isn’t exactly what a 13 year old girl is looking for but rather something that she will better appreciate over time.
@SouthernHope I know you’re from the south, where cash gifts are considered tacky, but cash is the appropriate gift for a bar/bat mitzvah. In multiples of 18, as others have said. It’s easy, and oh, so pleasy!
My kids have been to many bar/bat mitzvahs and alway gave cash. When my kids had their sweet 16, they received a lot Tiffany coins (not sure if they still have those) instead of cash. They pooled those coins together and got one thing they really wanted at Tiffany.
Amazon GC.
I gave cash to my son’s best friend. It’s fine - multiples of $18. They will be grateful. Really!
Last Bat Mitzvah my daughter went to, I gave $18 plus a gift. IT felt like a good compromise to me, as I’m uncomfortable with cash gifts, and was being urged by other parents to give $36 minimum (which is way more than we typically spend for a birthday gift for one of my children’s friends).
@Massmom i don’t live in the south anymore but I guess i do think of cash as the easy way out 
but I will consider as many have endorsed it! (and many thanks for the assortment of great ideas!)