Common Linux is hugely more heavy than WinMe, so all "light" versions would have to be damned improved to run properly.
Same for OpenOffice, which I ejected from my PIII 1.4GHz / 500MB for being soooo slow. Take an MS Office 97, it's perfect, fast, and has converters for more recent file formats.
W2k is slow. It accepts a smaller Ram than Xp does, for instance 500MB host W2k+antivirus+firewall+Firefox while Xp is too large for 500MB with these applications. But if you've enough Ram, Xp is a lot faster than W2k.
W2k has about nothing to do with WinMe: it inherits from Nt4, while WinMe inherits from W95-98-98se. WinMe is one of the fastest Windows ever, and I'd keep it preciously if it runs on this machine.
Internet Explorer 6.0sp1 is a usable browser and runs on WinMe. All Firefox v2 are said to run on WinMe. Opera v6 is said to run on a P1, Opera v9 recommends a PII and W98 at least, but is to run on W95-98-Nt4 if you provide Windows Installer.
The only brutal limit to WinMe and corresponding hardware is the lack of antivirus, if you connect to the Internet. For most other uses, older applications do the same job but faster.
Actually the light versions of linux which include: DSL, Feather linux, Ubuntulite, etc... Are made specifically for old hardware. They state the minimum hardware for Damnsmall linux is a 486 processor with 4 mb of ram.
Win2k can be a lot smaller than XP(Unless you are using a reduced size installer) and has fewer long term stability issues than ME. It also supports more functional anti-virus and browser products.
Linux can also run comfortably on 128 or 192 MB of ram with KDE2 or LXDE or a variety of other graphical interfaces. As other people have noted there are some extremely light distributions of linux, including fully functional distributions with Kernels smaller than 5MB