Android Browser that Saves and Fills Passwords
The standard Android browser does this for me on most (non-banking type) websites. When I start typing my username into the login box it shows a drop down of names used in that login box before. Selecting one of those fills the password box in too.

If you go into the browser and press Menu -> More -> Settings and then scroll down the settings window to the Privacy Settings section, there's a tick-box there for Remember Form Data, then if you scroll down a bit further to Security Settings, there's a tickbox for Remember Passwords. make sure that both of those are ticked and you should be ok.

I'm on a Galaxy S running the official Samsung Froyo 2.2 release, and I'm sure my old HTC Magic running stock Android Donut 1.6 did this too.