I think you are correct in so far as midnight is conventionally a term equivalent to 24.00hrs on a given date. So in that instance the cover is indeed from the start of Tuesday.
However, I suspect it's more to do with how your son has verbally expressed this to your wife and what he means is that he has arranged cover starting from 24.00hrs on Sunday. (or 00.01 hrs Monday )
Worth him checking though I'd have to agree.
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