Are you looking for web baesd stuff or something you can download / install?
To be honest a Quake III Arena always went down well when I was at school
In terms of puzzle stuff I sometimes play stuff on
addictinggames it's got lots of categories of games to choose from.
Not sure about high scores on them all but you could just have pupils take a screen shot of their score when they loose and submit it to a shared folder or something.