MEF certainly stands for Middle East Force, but I would be surprised to see it used on its own as part of a postal address. Maybe something "Maj T Smith, HQ MEF, BFPO 116" or similar. The MEF was spread over quite a large area including Palestine and, I believe, Iraq, so MEF on its own would have been fairly unhelpful.
If it was on a soldier's documents, that would make more sense especially if it was used in a general way, rather than for instance to denote a posting to a specific unit.