Rome in January

Recommend going to Ostia Antica and taking a side trio to Pompeii if possible. Also the archeology museum in Naples is fantastic.

I salute your 15-year-old for his/her interest in ancient history. I'm a history fanatic. I visited Rome in 1994 and again in 2022.

One idea is to visit the Jewish Ghetto of Rome. The main synagogue has an A-level museum, the temple itself features a superb design and there are multiple ancient Roman ruins in the neighborhood. The Jewish neighborhood hosts a part of the Teatro Marcello, which was the model for the actual Colosseum. I found the Teatro Marcello more interesting than the Colosseum. Go here: https://anamericaninrome.com/2017/04/teatro-marcello-rome/

Ostia Antica, the ancient port of Rome, is worth visiting. I saw it on my 2022 trip: https://www.ostia-antica.org/

The Appian Way (https://www.ostrichtrails.com/europe/italy/appian-way-walk/) is an excellent walk or bicycle ride. The Ancient Romans built the road, and it's still there. It's like walking through history with the monuments, ancient structures and aqueducts. I walked the Appian to Aqueduct Park (https://www.romesightseeing.net/aqueduct-park/). If you bicycle the Appian, be extra careful: I saw a woman who had tumbled from her bicycle. You would be bicycling over 2,000-year-old pavement.

The Christian Catacombs are definitely worth visiting (and they're near the Appian Way): https://www.catacombesancallisto.it/en/index.php

The Vatican Museum and Sistine Chapel (and St. Peter's Basilica) are excellent sites.

If you really want to geek out on history -- I certainly did -- you could visit Pompeii and come back in a day. I did that in 2022. However, if you went there and back to Rome in a single day, you'd probably miss the Naples Archeological Museum, which holds the artifacts from Pompeii. Also, Naples is a pretty cool city. (Keep in mind that Pompeii is about 20 miles outside of Naples, but you can get to Pompeii via a bus or transit or splurge on an Uber.)

Yet one more idea: You could tour some of the breathtaking churches in Rome's center. Some I saw: The Basilica di Santa Maria Maggiore, San Giovanni in Laterano (the first Vatican), Santa Maria in Trastevere, Santa Prassede & San Zeno Chapel (probably the most amazing of the churches and some tourists miss Prassede) and San Clemente.

You didn't say how many days you have, or if you are also visiting other locations in Italy besides Rome. That information will help narrow down the recommendations.
Given the interest in history and archaeology, I'd go all in on the Colosseum and Forum and probably the super sites, as well. You'll need to read the descriptions and decide. It can be overwhelming!
The Capitoline Museum is an underappreciated gem, in my opinion, and it doesn't take a ton of time to go through.
We've been to Rome several times and still haven't made it to Castel Sant'Angelo but it remains high on my list to visit.
Church of San Clemente is very interesting from an archaeological perspective - layers of previous rooms have been uncovered. A must see, imo.

You’re right, that's a big piece of missing info! We’ve got 3 full days in Rome, but will be in Lucca (planning day trips from there to Florence, Pisa, Bologna?) for 9 days before that.

We visited Ostia Antica for the 2nd time last January (2024) and although it was great, we were bummed that all of the mosaics were covered with tarps for the winter --- still well worth going there, but I just wanted to warn you.

Capitoline museum is wonderful, and the view of the Forum from its balcony means that the Forum makes sense in a way that it doesn't when you are actually in it. I hope the gravestone epitaphs in the underground passageway between the two parts of the museum are still there --- reading these was one of the best things we have done in our approximately 400 days spent in Italy so far.

I really enjoyed the Baths of Caracalla --- they help a bit to make you feel how huge all these ruined buildings were when intact. Also the Pantheon, of course --- still intact!