That's an interesting little video, although the poster's choice of background music is.... strange, to say the least. The guitar instrumental almost works (anybody know who that is?)... but it would be another 6 or 7 years from the time of Project Mercury for "Daydream Believer" to hit the charts.
Interesting side note though, while made popular by The Monkees, that song was actually written by former Kingston Trio member, John Stewart- who sadly died last January. John Stewart was a good friend of Scott Carpenter's- as a matter of fact he was the entertainment for the Astronaut Hall Of Fame gala banquet a few years ago.
So in that way, I guess it's distantly related to Project Mercury after all.