1590s, "to assign quarters to, to direct (a soldier) by note to a lodging place," from a noun meaning "a ticket given by a military officer directing a person to whom it is addressed to provide board and lodging for the soldier carrying it" (1640s). This was a specific use of the word, which earlier meant merely "official record or register" (late 13c.), from Anglo-French billette "list, schedule," diminutive of bille "written statement" (see bill (n.1)) with -let. From 1830 in the sense "place where a soldier is lodged." Related: Billeted; billeting.
"short, thick stick of wood used for fuel," mid-15c., from Middle French billette, diminutive of bille "stick of wood," from Medieval Latin billia "tree, trunk," which is possibly from Gaulish (compare Irish bile "tree trunk").