Is the error message the same as before (Invalid password)? If so, you're probably right, the logs won't likely tell us anything more. Do you have any characters in the Rdio username and/or password that are NOT letters/numbers or which are not standard English characters? Assuming not, to debug this further I think you'll have to actually edit the plugin source a bit yourself to add some extra debug info. Could you do the following for me:
1. FInd the following file and open it in a text editor: <xbmc_home>/addons/plugin.audio.rdio/rdioxbmc.py
2. Find the following line:
self._addon.log_notice("Logging in to Rdio using URL %s" % (login_url))
and add the following line below it:
self._addon.log_notice("Logging in to Rdio using username %s and password %s" % (username, password))
3. Try to log in again through the plugin
4. Have a look at the XBMC log file and find the log line above with the username and password - verify that these are correct. Do NOT post this log to the forum!
5. Be sure to remove the line that you added in rdioxbmc.py so that you don't accidentally post your password to the forums in the future
This will verify that the username/password are actually being read properly from the XBMC settigns. Let me know the results. Based on the results we'll try a few other things after.