Just a heads up. Internet download speed is not onerous but it is the UPLOAD speed which is the real bottleneck for a home private server.
Also, as you won't be home all the time and your son/daughter will be home more so, you need to have a turn key system for them to restart the server in case you are not around. They need to be able to identify your IP address to communicate to their friends.
And you need to set a time limit every night so that you may do necessary system backups. Yeah they won't be happy about that early bed time but on the odd occasion you are asked to do a roll back - they will be truly thankful!
Mostly they'll play cooperatively as they are acutely understanding that they and they alone are plotting the destiny of THEIR world and no stranger on some remote internet server can take that away from them. But... disagreements occur and sadly you may be required to mediate to prevent unfair exclusions and to, alas, do the odd roll back when someone goes crazy with TNT.