Well, I give up. I am using an ibook G4 with Leopard and Mac Mail. I have two gmail accounts set up on my ibook, one for me and one for my wife. Both have worked flawlessly through Gmail. Recently, although my wife's account can receive emails, she cannot sent out email as 'can't use this server' message comes up and it appears at that point that her account is offline. All accounts are online according to that option in the Mail menu. Both accounts are set up exactly the same using the same gmail outgoing server smtp.gmail.com:gmail email address. If I change the out going server to mine ie,smtp.gmail.com:my gmail email address then the message will go.(although any relies will come to me rather than her). What have I done? Repaired permissions, used Onyx, deleted the mail prefs and re-entered new email accounts for both of us (actually "started all over again").
Her account works fine via her iPhone so the problem seems to be with my iBook in some manner..