Maybe the cleaners are going in and downloading teh internet?!
I was thinking something along those lines, maybe somebody is scheduling downloads via bittorrent or something... but that wouldn't make NICs switch to a different setting.
It does sound strange. Why not try leaving on the other wired computers and log into them and see if the same thing happens, it might be the switched ports on the router changing... maybe change around the cables to troubleshoot it further? Maybe its LogMeIn thats conflicting with the card when the connection initiates. Try installing a free VNC server as well... I use UltraVNC with either a VPN connection or port-forwarding on the router.