I'm not positive, as I'm going from memory, but you should be able to
get Thunderbird to sync your mail with Palm's default Mail app (PalmOS
4.x, not sure about PalmOS 5 or PalmOS 3).
The only oddities about it are that:
1) You need to tell the Palm mail conduit to use the Eudora conduit
(and not the Netscape one).
2) You MAY need to install the Thunderbird Offline extension that lets
you put Thunderbird into an off-line mode.