The surname Bui and its rare variant Buy present an uncertain origin, with several possible interpretations of anthroponymic and occupational type. According to E. Rossoni, they could derive from an abbreviation of the Celtic name Buic, from forms with the loss of the initial syllable of names such as the Frankish Lebuin, or from variants of the Lombard name Albuin. These hypotheses link it with the tradition of Germanic and Celtic names spread in medieval Italy.

From the geographical point of view, E. Rossoni notes that Bui presents clusters in various areas of the Italian centre-north, such as between Bergamo and Brescia, in Ferrara, between Siena and Perugia, and also in the Rome area, while Buy is a practically unique form.

For his part, F. De Laurentiis proposes a different interpretation, linking the surname to nicknames and trade names derived from bove ("ox") and its dialectal variants (bovaro, boaro), which suggests a possible occupational origin linked to work with cattle.

Taken together, the various interpretations agree in pointing to a multiple and non-univocal origin, where anthroponymic roots of Germanic or Celtic tradition may converge with possible nickname-based or occupational derivations linked to the rural world.