(Anglo-Saxon). Chamber; originally applied to a mansion, like the French hotel. Hence Clifford's Inn, once the mansion of De Clifford; Lincoln's Inn, the mansion of the Earls of Lincoln; Gray's Inn, that of the Lords Gray, etc.
Now, whenas Phoebus, with his fiery waine. Unto his inne began to draw a pace.
Source: Dictionary of Phrase and Fable, E. Cobham Brewer, 1894