The rabbit (토끼, tokki) is read through cleverness and quick thinking more than raw power — Korean folklore has a long-running soft spot for the rabbit as the underdog who wins through wit, most famously in 별주부전 (the story of the rabbit and the turtle), where the rabbit repeatedly outsmarts a far more powerful opponent.
There's also a lasting lunar connection worth knowing: in Korean folklore, the shapes on the moon's surface are traditionally read as a rabbit pounding rice cake (떡) under a tree, an image still referenced constantly in children's stories and Chuseok imagery. A rabbit dream that also involves the moon leans this reading toward family, home, and domestic good fortune rather than pure cleverness.
Catching a rabbit is read as a small, quick gain won through cleverness rather than force — outsmarting a situation rather than overpowering it. A rabbit escaping your grasp is read as a missed opportunity specifically because you were outmaneuvered, a gentler, less alarming version of the same "opportunity slipping away" reading that shows up with other animals.
A rabbit running calmly rather than fleeing in fear is read favorably, often as good news arriving quickly and without complication — speed here reads as luck rather than danger, unlike a chase-dream reading elsewhere on this site.