You might find this Wikipedia article to be interesting and informative.
Microsoft started development on the .NET Framework in the late 1990s originally under the name of Next Generation Windows Services (NGWS). By late 2000 the first beta versions of .NET 1.0 were released.
An old press release for the .NET family alludes to its previous title of Next Generation Windows Services (NGWS). If sarcasm is more your cup of tea, this announcement from The Register is interesting as well.
And according to this Wikipedia article on Microsoft codenames, it appears that .NET/NGWS went by the names Lightning and Project 42.
Project Lightning was the original codename for the Common Language Runtime in 1997. The team was based in building 42, hence Project 42. "Next Generation Windows Services" appeared in the earliest press releases about the upcoming platform.
Wikipedia links to an interview of Jay Roxe and an article from The Age as evidence for this information.
Jay tells us that development had begun in earnest at least by 1997, as that's when he joined the team:
OK, well let me give you the history. I joined what is now the .NET Framework team, or the Common Language Runtime team, back in November of 1997. [This was] back when it was called Project Lightning, then it became COM+, then it became Project 42, then we had this nice little re-org that made it Project 21 ? we lost half the team.
And so, I wrote things like String and StringBulder, and I wrote the initial implementation, although I did not own it forever, all of the base types like Int [16, 32, and 64], and double, and all of those. I did some of the work on Object and was Dev Lead for the System.IO classes, the globalization, and a bunch of the collections work as well.
A blog post by Jason Zander on an unrelated topic gives us the interesting tidbit of information that the "Lightning" codename was chosen by the founder of the CLR team, Mike Toutonghi:
The original name of the CLR team (chosen by team founder and former Microsoft Distinguished Engineer Mike Toutonghi) was "Lighting". Larry Sullivan's dev team created an ntsd extension dll to help facilitate the bootstrapping of v1.0. We called it strike.dll (get it? "Lightning Strike"? yeah, I know, ba'dump bum).
And James Kovacs's C#/.NET History Lesson fills in a few more of the gaps. This Stack Overflow question is also worth a read, for those interested in history.