Yours, Etc.: Origins and Uses of 8 Sign-Offs
late 14c., from amidde (c.1200), from Old English on middan "in the middle," from dative singular of midde "mid, middle" (see middle); the phrase evidently was felt as "in (the) middle" and thus followed by a genitive case, and if this had endured we would follow it today with of. (See amidst for further evolution along this line).
The same applies to equivalents in Latin (in medio) and Greek (en meso), both originally adjective phrases which evolved to take the genitive case. But in later Old English on middan also was treated as a preposition and followed by dative. Used in compounds from early 13c. (e.g. amidships, attested from 1690s and retaining the genitive, as the compounds usually did in early Middle English, suggesting this one is considerably older than the written record of it.)