Okay, so scratch all of the above. It actually doesn't work to get the file to play on the native Music App AND it's too involved AND involves using the computer.
Here is an easier way that I have totally checked that it works and you can do it all from your phone. You must be comfortable with iFile ...
Programs needed:
Pandora Downloader Tweak
Pandora
iFile
PwnTunes (free trial offer)
1. Record the song you want with pDownloader. Download goes to var/mobile/Media/Pandora
2. In iFile, navigate to var/mobile/Media/Pandora and cut the song that you want to play in Music App
3. While still in iFile, paste the song into var/mobile/Media/My Music (which is the directory created by PwnTunes)
4. From Settings, open PwnTunes and turn Needs Sync 'On'
5. Open the native Music App and it will automatically import the song(s) you just moved over.
6. Look for your song in songs because there will be no album, artist, etc. There will also be no cover art, but the song is there!
7. Golden!
Good Luck and this does work. Seems involved but once you get into it, it goes really fast especially if you set a bookmark in iFile!
Scarlet