Uncertain. Perhaps a denominal verb, for which compare Old Uyghurبوى(boy, “paint, color”).
This etymology is missing or incomplete. Please add to it, or discuss it at the Etymology scriptorium. Particularly: “Old Uyghur term above is unreliable for an etymology (see that entry). As per Tietze's Dictionary (Volume I, 2016, page 770), the aberrant relation between *bodu- and *bodug is seen also with *āčïg(“bitter, sour”) <> *āčï-(“to become bitter, sour”), so it is not an isolated case. More research is needed for this page.”
^ Sanžejev, G. D., Orlovskaja, M. N., Ševernina, Z. V. (2015) Etimologičeskij slovarʹ mongolʹskix jazykov: v 3 t. (in Russian), volume I, Moscow: Institute of Oriental Studies of the Russian Academy of Sciences, page 93
Clauson, Gerard (1972) “boḏu:-”, in An Etymological Dictionary of pre-thirteenth-century Turkish, Oxford: Clarendon Press, page 300
Räsänen, Martti (1969) Versuch eines etymologischen Wörterbuchs der Türksprachen (in German), Helsinki: Suomalais-ugrilainen seura, page 77
Sevortjan, E. V. (1978) Etimologičeskij slovarʹ tjurkskix jazykov (in Russian), volume II, Moscow: Nauka, page 178
Starostin, Sergei, Dybo, Anna, Mudrak, Oleg (2003) “644”, in Etymological dictionary of the Altaic languages (Handbuch der Orientalistik; VIII.8), Leiden, New York, Köln: E.J. Brill