鸟 (bird) is the indexing radical, immediately flagging this as a bird name. 鸠 covers turtledoves and pigeon-like birds, joining the huge bird family of 鸡 (chicken), 鸭 (duck), and 鸽 (pigeon).
九 supplies the sound, drifted slightly to jiū. The choice was almost certainly onomatopoeic too - the cooing of doves and turtledoves sounds a bit like jiu-jiu. The same phonetic feeds 究 (investigate) and 仇 (enemy).
