Amsterdam in January

I'd go for the day after xmas to after the New Year. There is a lively New Years eve in Amsterdam.
Otherwise, things are lot less expensive lodging wise near the center and a lot less crowded after the New Year.
I do not know if Xmas markets matter or is something that a 21 year old would find enjoyable in a wild place like Amsterdam.
I would let them make their own decisions based on their own research. I travel with my nieces and nephews that are about the same age and let them decide and they are generally happy with that

As a bonus I believe Amsterdam also has their evening light festival in January. Along the canals, bridges and buildings are lit up with beautiful and artistic projections.

ditto on the advice. You can make recommendations, but let them choose their sights. If you want to be involved, offer to make their reservations once they have made their selections - with their credit card of course :-)

Yes you'd be surprised what youngsters chose to do that you would never have thought about.
And they know how to get places with those telephones that they all carry

Amsterdam isn't as famous for Christmas Markets as Germany. But there are still some good ones nearby. I love the Europe Christmas markets. There are festive, fun, and get you into the holiday season. If you love Christmas then I would visit in December. But keep in mind that there is a chance that of the museums and attractions would be closed for the holiday season.

You just have to look up each place that you want to go to on the web to see the hours. Most things are open except on xmas and new years day. The Ann Frank House is open on Xmas and New years day

The Rijks Museum is open 365 days a year, so it will be open on xmas and New Years.

Make sure to take layers. Amsterdam - January = cold and windy