I currently have a 701 4G, a 901 20G and have a tracking number for the brand spanking 1000 HE.
Yep, I love dem ASUS!
The question was: which distro for an EeePC. I have tried ubuntu-eee (now easy peasy), xandros, debian eee, cruncheee and a plethora of small and variously customised distros. I guess I would recommend either eeebuntu or debian eee and leave the decision up to you. It boils down to your preference: ubuntu or debian, as I have found the eee forks of these distros to be roughly equivalent.
I have settled on dual booting the original ASUS xandros with eeebuntu (base). The quick off-to-google time is the only real reason I keep the xandros OS on, although I do like to keep the ASUS software up to date simply as a point of comparison. Eeebuntu worked OOTB for me. The base version is lean, and allows you to install the programs you need without fear of fluff. One thing for eeebuntu: for some reason, the downthemall! addon for firefox came installed as admin. This may be fixed now, but its a good thing to be aware of; if you run ffox for the first time as gksu (not sudo!) you can decline the license agreement and the addon is uninstalled.
I chose ubuntu 'cos I like brown, I've been using it for years, and the volume of forum help is pretty awesome if you run into trouble.
Dual booting is a good option in my books. Perhaps you could dual boot windows 7 with your chosen linux distro?
Finally, I just wanted to plug zareason.com. They sell ubuntu based machines, including the eee 1000, that have been thoroughly road tested and come with 2 years of support.
Its the kind of business that got my mum off micro$oft.