Well, assuming a Redstone w/boosters could put a capsule into orbit (which I seriously doubt), the Mercury wasn't able to change its orbit or maneuver to rendezvous.

That said, I've got a real soft-spot for that sort of story. Sort of a Rocket Ship Galileo thing. I remember an after-school TV movie from when I was a kid was called Stowaway to the Moon: A kid hides in the trash compartment of an Apollo capsule the night before launch. The crew finds him shortly after TLI, so they're committed to the mission. While the LM crew was on the lunar surface, the Command Module Pilot gets incapacitatingly sick. Of course, the kid (with a lot of coaching from the ground) saves the day. That really had my head in the clouds for a while.

I drew the line at Space Camp, however.

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"Transport of the mails, transport of the human voice, transport of flickering pictures - in this century, as in others, our highest accomplishments still have the single aim of bringing men together." St. Exupery