The story
The noun is derived from bit (“small amount of something; smallest unit of storage in a digital computer, consisting of a binary digit”) + bucket. Bit in this context originally referred to small pieces of paper punched out from paper tape or punch cards (see sense 1), but came to be regarded as the unit of data storage (sense 2). The verb is derived from the noun.
Etymology from Wiktionary, CC BY-SA 4.0.