I am not sure if you are looking for “solo accompaniment” ie playing and singing or just the “solo/break” part of Girl of the North Country, where he does that Allman Bros thing. Anyway, for just struming and singing Joe Walsh’s (not the Eagle, the Gibson Brother guy) Mole in the Ground works well as a solo Mando piece.
As for just keeping the “chops” for solo breaks, fiddle tunes work great for me. Either Bens stuff here or basic tunes (I use the fiddlers fake book).
For someting to aspire to/drool over/pull your hair out to - solo mandolin - a fiddle tune “wildwood flower” into “going steady” http://www.youtube.com/watch?v=ZTis0S9zxXg