A steady old horse, a child's horse. Dobby, a silly old man. Dobbies, house-elves similar to brownies. All these are one and the same word. The dobbies lived in the house, were very thin and shaggy, very kind to servants and children, and did many a little service when people had their hands full.
“Sober Dobbin lifts his clumsy heel.”
Source: Dictionary of Phrase and Fable, E. Cobham Brewer, 1894