Get your kid used to drinking more water. I used to get nosebleeds all the time in winter as a kid and teenager, but I've since been a very avid water drinker and my rate of nosebleeds has gone down considerably.

I have been looking into some whole home humidifiers myself, because it gets so dry in the winter where I am at. Not sure they are an option for you, but they hook right into the HVAC. That way ventilation would help rather than hurt.

There are energy recovery wheels, which take advantage of some sort of adsorbent to transfer humidity from the outbound to inbound stream and recover some of it (or the reverse, keep indoor air conditioned)

Put plants on every windowsill?

Steam heat in the winter and no A/C in the summer. It cuts down on dust too.

(yes I know this is not a practical solution)

indoor plants would provide humidity and some co2 decrease (not sure how much)

Ayr Saline Nasal Gel.

Close the vent in his room?

Small turnover gives large wins.

There are heat-exchanging air exchangers for high-efficiency construction. I'm not aware of (and don't find) humidity exchanges, though they may exist.

Hung dampened linens (towels, bedsheets) can make a big difference. Many HVAC systems offer high-efficiency humidifiers, and the old standbye of having a kettle on the stove can work (with an energy cost).

Humidity in general is fairly cheaply acquired. Dehumidifying moist air takes much more energy.