This character was found in oracle bone scripts depicting a sacrifice (a human or animal body) that is cut in half. This kind of practice mainly happened in Shang dynasty and was gradually eliminated after Zhou's conquest of Shang, leading to obscurity of the character's original meaning. An old variant is 戼. According to Shuowen Jiezi, it is an open door, like 門／𨳇 (mén) but reversed.
Derived from root 流 (OC *r(j)u) with volitional or agentive OC prefix *m- (see Sagart, 1999); so *m-ruu means "(cause to flow >) pour out, empty" > *m-ruuʔ 卯 "pouring or emptying stage" i.e. "waning moon", "with the word in its sacrificial applications meaning 'to blood-let'" (Smith, 2011).
Association with the rabbit was probably arbitrary, just as how 辰 was arbitrarily associated with the dragon (see Ferlus, 2013).