Homer uses both μένω and μίμνω, the latter of which looks to me like a reduplicated form. Wiktionary gives definitions that seem almost identical, and says that μένω supplies all of the tenses of μίμνω except the present and imperfect. Cunliffe also gives almost identical definitions. Why, then, do we have the two different forms? Is there a different shade of meaning? Are they from different dialects? Is one more archaic?
You're right that μίμνω is a reduplicated form of μένω. A plurality of attestations seem to come from the epics, and L&S call it a poetic reduplication. μένω was in much wider use, and seems to have been the unmarked form. There's actually a further enlarged form μιμνάζω, which again is mostly limited to the epics. All three verbs mean the same thing, and it's only the register that differs.
(Not sure what Aretaeus' deal was.)
I suspect μίμνω is the oldest form, and was a regular reduplicated present (imperfective) during the Proto-Indo-European stage. The aorist wouldn't have had the reduplication, and as reduplicated presents ceased to be productive over time, the non-reduplicated stem would have spread to the present by analogy. Archaic oral poetry preserved the now-irregular form.
It's actually kind of unhelpful that we think of the present stem as the "basic" verb stem. In many ways it makes more sense to think of the aorist as the unmarked stem, as it's fundamentally from this that the other stems are derived; for the present, this sometimes happens by adding a nasal infix (e.g. λαμβαν- from λαβ-) or the so-called inchoative -σκ- suffix (εὐρισκ- from εὐρ-), sometimes by i-reduplication (γίγν- from γεν-), sometimes by combinations of these things (γιγνωσκ- from γνω-), sometimes by none. (The development of the sigmatic aorist admittedly makes this picture slightly messier.)
If you think of μεν- as one verb that happens to have two possible ways to form the present, as the Greeks themselves would have done, rather than as being two different verbs that happen to share most of their morphology, it looks a lot less unusual.