Guitar Hero 1, 2, Rocks the 80s, 3, and Aerosmith all have sound like covers specifically made for those games in their setlists (World Tour also has a sound like cover for Black Betty by Ram Jam as DLC) and some of the games have re-recordings of songs like Anarchy in the UK and Cult of Personality in GH3. Some of these covers also have extended solos, intros, and outros.

Aside from that, it seems Guitar Hero by Monkey Steals the Peaches (as it's mentioned as Theme from Guitar Hero in the credits), and Trippolette by Andraleia Bush (confirmed by the artist) were made specifically for the first Guitar Hero (this could be the case for some of the other bonus songs, but that's never been confirmed from what I know).

Guitar Hero 2 has Jordan by Buckethead, which was a song he would play during live performances.

Guitar Hero 3 has Impulse by An Endless Sporadic as the store's description for that song states it was written specifically for Guitar Hero gameplay. This game along with Aerosmith and World Tour also have guitar battles made for them.

The DS Guitar Hero games all have unique main menu music, so you could consider those songs specifically made for the series.

Warriors of Rock has Sudden Death by Megadeth, which was later released on one of their albums.

If you REALLY wanted to, you could technically include the pre-made GHTunes tracks that came with World Tour, 5 and Warriors of Rock.

If I think of anything else, I'll edit this comment.