(All of these opinions come from an older teen who has been visiting Broadway since she was thirteen)
The Play That Goes Wrong isn't a musical, but it's hilarious and I couldn't imagine a thirteen-year-old who wouldn't like it. It's all slapstick and cheap humor – in a good way – and very enjoyable.
Waitress should generally be fine, but be aware that it deals with subjects such as extramarital affairs, and has some allusions to sex (there's one montage that gets fairly graphic, but it's done hyperbolic, so it could fly over her head), so it really depends on how mature she is.
The Great Comet of 1812 is an amazing spectacle, and my favorite show on Broadway right now. The music isn't super easy to get into, but there's so much pizazz that it makes up for it. Like Waitress, there's an affair, but everyone who gets involved in it faces repercussions and is even offered forgiveness at the end after apologizing for their actions, giving it a much more positive message than the former, imo.
Bandstand deals a lot with PTSD and some heavy stuff, but I honestly don't know enough about it to tell you about its appropriateness.
Miss Saigon is a great show, but definitely not appropriate for a child. The first act largely takes place in a strip club, takes place during the Vietnam War, and ends with the very lovable, teenaged female protagonist committing suicide. There's a lot of clearly implied sex, and while the overall message is a good one (it's largely about familial love and self-sacrifice), I don't think it's a good idea to take anyone under fifteen or sixteen to the show unless they're quite mature.
Beautiful again deals with an affair, but is generally quite vanilla, however I couldn't see a thirteen-year-old getting into it, as it is a musical about Carole King. Teenage girls aren't exactly the target audience.
Groundhog Day I haven't seen, but I believe that there's a suicide in the show. I'm not sure how graphic it is, however. But from my understanding it's been marketed as 13+, so I can assume it isn't that bad. Otherwise, I hear it's quite fun!
Kinky Boots isn't graphic beyond having a character who is a drag queen, but depending on your family's liberality/conservatism, that might not be graphic, but I don't know your personal beliefs so I can't really speak for them, haha. Otherwise nothing strikes me as inappropriate about it, however just from a personal standpoint I despise the music.
Once on This Island isn't graphic at all, but it's not coming out for quite some time. I could see a thirteen-year-old enjoying it, but I could see her enjoying other shows much more.
War Paint and Hello, Dolly! aren't inappropriate, but again, I don't think they're the types of shows your daughter would really be interested in. The scores are very old school and none of the characters are very relatable to a teen audience.
Sunset Boulevard is one of my favorites, but I could see a younger girl getting bored with it for the same reasons I mentioned for War Paint/Hello, Dolly! Also, the protagonist gets shot at the end (it happens onstage but there's no blood or anything, he just staggers and falls into the "pool"
, and the show centers around a younger man (Michael Xavier is 39) essentially prostituting himself to a significantly older woman (I believe Glenn Close is pushing seventy) in exchange for living in her mansion and having all of his worldly needs met.
All in all, I'd suggest The Great Comet and The Play That Goes Wrong the most out of this bunch, but for a thirteen-year-old I would suggest Anastasia or School of Rock over them, just because they're extremely PG and fun, despite both not being as good shows as the formers.
Hope this helped!