на честном слове is an idiom. Эта машина едет на честном слове, это работает на честном слове. This means that this plane flies defying all physics laws (well, just because someone had promiced it would fly).
Aha, so it's basically a loose translation of "on a prayer". To avoid sounding so religious, you could also say something like The plane seemed to be flying on the sheer willpower (or "determination") of the crew.