Green’s Dictionary of Slang

(US) the penis.

1960		Reuss & Legman Songs Mother Never Taught Me n.p.: If all them young ladies wore a tiny bikini / I’d walk around with a three-foot bodini [HDAS].
1980		E. McDowell Our Honor 81: Till I unzipped my fly / When she saw my budini / I thought she would die [HDAS].