London is so replete with history that you can’t walk down a block without seeing something historic, so even if she wants to avoid it, history will be all around her! :-)
That said, try Shoreditch and the area around Spitalfields and Brick Lane for a young, modern vibe, including lots of really cool street art. It’s funky and vibrant, not stuffy and “historic.” The Spitalfields Market has excellent stalls with independent artists, fashion, wearable art, etc., plus some excellent food. https://www.spitalfields.co.uk/
The Saatchi Gallery has super exhibits, and concentrates on modern works. https://www.saatchigallery.com/ If you visit you’ll be on the King’s Road near Sloane Square. If she has any interest in fashion/boutiques/shopping, it’s a great area.
The Tate Modern features modern art, and it’s great. https://www.tate.org.uk/visit/tate-modern
If you visit the Tate you’ll be on the south bank, so you can visit Shakespeare’s Globe. Catch a show as a groundling - I bet she’d love that! There are tours, if she’d like a peek backstage. https://www.shakespearesglobe.com/whats-on/pinocchio/
Camden Town has a youthy vibe. https://camdenstablesmarket.com/ Tons of fun shops and edgy merch. Stables Market is great for teens. There’s great shopping there - if she’s into thrifting, this would be right up her alley.
There’s interactive theatre (Squid Game, Fawlty Towers) if that sounds attractive to her. Otherwise, the sky is the limit when it comes to shows… big stuff in the West End, plus new plays, experimental theatre, and the like in smaller venues.
I think London is perfect for a 16 year old. I visited the first time when I was 15, and vowed that I would live there someday. And now I do! :-)
Hope you have a blast!