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Zaimi could be Greek, though Zaimis is the regular Greek version. Zaimi is also Albanian. I think it's safe to say that, whether Greek or Albanian the name will have the same meaning.
The -i ending is probably a patronymic ending, as in Albanian surnames such Mehmeti and Kjelmendi. As for the meaning of Zaim, possibly from the Arabic Za'im, via the Turkish, dating back to the time when the Balkans were part of the Ottoman Empire. The Arabic Za'im is a title, meaning something like 'leader' or 'chief'.
The above is just speculation on the meaning of the name, no concrete evidence.