Well, I have to say that I use youtube for my band videos. The limit is 10 minutes or 100gb whichever is less. Also, if they don't want to share stuff with the world, they can set the account to private status and then the account owner has to authorize individual youtube accounts to view the videos.
That being said, video can be kind of a complicated thing. Youtube hosts stuff in local data centers all over the world so you get that advantage for DL speeds.
However, the DL speed is vastly more effected by the work that is done to prepare a video for hosting. If your relative is using a consumer level camera that records to DVD, the DL speed is going to suck a tennis ball through a garden hose if that is uploaded directly to anything like an FTP host or if they use bittorrent.
Youtube will convert the upload to a much better format but you may have to wait an hour after the upload for the conversion process to go through. So you get the good speed but don't call people on the phone to tell them that you just uploaded new stuff. Also, youtube may not use the best possible settings for the conversion.
Honestly SSS, I am not getting a good read on how technical to go here but if they do not yet have a camera I would recommend anything that records to miniDV tapes. If you plug the camera into a default install of XP, windows movie maker will just come up automatically and they can capture the whole tape to .wmv in a single pass.
It also bears noting that movie maker is a pain in the butt to edit clips out of and are also .wmv is one of the formats that put you into waiting for youtube to allow you to tell people when something is ready.
I can also recommend software if you think that that is reasonable but for brevity, let me stop now.