Uncertain. Possibly deverbal from tillen (“to lift up”); some early attestations of the word refer specifically to the hatch used to close off a dovecote. The meaning "dovecote" then originates as a pars pro toto usage.
An alternative possibility is that the word represents a Frisian form corresponding to deel (“plank”) (referring to a plank on which doves are kept), ultimately from Proto-Germanic *þiljǭ.