The story
From bread + board. Regarding definition 3: the etymology is either that electric circuits used to be prototyped on actual bread boards (definition 1) populated with nails, or that solder breadboards have lots of holes and are usually brown (they visually resemble a slice of bread), or a merging of the two. The etymology is less obvious today given the popularity of solderless breadboards.
Etymology from Wiktionary, CC BY-SA 4.0.