So, assuming your player chooses Wizard as their Magic Initiate class:
If their chosen Initiate 1st-level spell is Mage Armor, they learn the spell (add it to their spellbook) and also can cast it for free once a day. However, to cast it with spell slots, they must also prepare it as one of their spells for the day. This does allow them to pick other spells for their starting spells known, and still have the ability to prepare Mage Armor if they wish.
However, if they already know Mage Armor, taking the feat does not automatically allow them to learn a different spell for free; if they did, they would cast that one 1/day instead of Mage Armor.