We may be left guessing for some time - my guess is what you said upstring: Earhart may have wanted a way to energize the mic for speaking without having to hold the button down manually.
This whole approach of a 'switch panel' appears to be a task-saving device to improve cockpit handling needs. She was solo, for practical purposes, in a fair-sized transport with duties that might get busy easily enough. It might also be seen as a way to reduce the chance of error in radio handling - which wouldn't have hurt her (and may not have been a strong enough measure...).
Anyway, "MIKE" ("MIC") / "OFF" gives two basic functions - the implied "on" probably relating to one of two things:
1 - Arming the mic (not 'keying' the mic) - which seems redundant to the "C.W. / PHONE" switch, not to mention hardly necessary, and
2 - Keying the mic.
My thought is guess number 2, that she whacked the mic "on" when ready to talk, and talked away until done, then flipped it "off" to let the battery recover, etc. That might well have also saved the chance of problems with a semi-reliable mic button / switch on the mic itself in the day - and avoid the ambiguity of whether the mic stayed keyed as intended or not, etc.
It also happens to jibe somewhat with the long transmission cycles Betty reported - as in, "Fred, now you sit down - we're gonna kill the batteries if I don't get back to the mic, make sense, and get off for a while...".