Likely a babble word; compare dialectal German Bübbi (“teat”). Some older references connected the word to French poupe, but this is considered "very doubtful" by the OED. Compare Latin pūpa (“little girl”).
Meanings
(dated, slang) A woman's breast.
Mr. Douglas got up from the couch, shucked down his pants, and then knelt down beside my mother. He reached out and grabbed her big round bubbies, and began to squeeze and play with them, while he teased her by prodding his cock against her red behind. 2009, Arlene Gorey, My Spanking Diary:
Chlo: What do you mean (uncivil as you are) / To touch my breaſts and leave my boſome bare? / Daph: Theſe pretty bubbies firſt I make my own. 1685, John Dryden, Sylvae:
Perhaps from brother, as pronounced by young children who are not yet able to properly pronounce its complex consonants, but alternatively perhaps from Pennsylvania German; see bub. Compare sissy.
Meanings
(childish) Familiar term of address for a boy; bub; bubba.
Mother sent me to hunt you; she is dying, and sissy and bubby are hungry and the baby is crying, and we're afraid the cross man will come. 1862, Caroline Augusta White Soule, Wine or water; a tale of New England, →OCLC, page 78: