Mark Pilgrim expounds on the history of the tilde.
One of the unanswered questions was "When did it migrate into the world of web servers to provide a cheap and simple way of giving individual users their own web sites?"
You could just email Rob McCool or someone and ask them when this was introduced, but it's much more fun to play archaeologist, isn't it?
The CERN/W3C web server added support for ~username in 1994. Change notes (sorry, no direct link to the relevant version, they don't have anchors on that page) for version 2.15beta, released 11 February 1994, say this: "User-supported directories enabling URLs starting with /~username".
NCSA's web server is more difficult. The version history page first mention of the feature, I think, at version 1.0a4, but that was already a bug fix. It doesn't mention release dates, either. Grabbing the old versions from their ftp server, the feature wasn't in 0.5 but it was in 1.1 (obviously), which was apparently released somewhere around March 25th, 1994. At least that's the date of the newest files. Sadly none of the versions between those two are available for download.
Quick Googling didn't find version history for Plexus.
Switching to Google Groups,Article page