There could be any number of reasons for the lagging and high pings.
Wifi is inherently slower than Ethernet. And wireless is quite subject to interference from multiple sources in the surrounding environment.
However, if there have been no network problems to date then his Xbox and internet use is simply pushing network traffic to the point of congestion. Or the additional traffic has exposed some flaw or problem with your network. Could be physical or configuration.
And if his system/Xbox is not correctly configured then that may also be a contributing factor.
Does not matter that you have a "decent modem and internet". And router for that matter.....
He could try relocating his Xbox higher or elsewhere in his/their room. Double check that the Xbox is correctly configured for your wireless network.
However, the ping may even be a problem via your ISP or elsewhere. Even the gaming site....
Overall, the real issues are not technical. If he is not paying the bills, then he has no say in the matter.
Strongly recommend a "family meeting" to work out the terms and responsibilities of the relationships involved.
Otherwise will not end well for anyone.....