In ancient Hebrew the "i" (yod) is added to denote first person possessive as in "abi" and "immi" (my father and my mother). It appears that possession implies intimacy in a relationship and thus a way to make a name more precious and personal.
Refer to a Hebrew language primer, for example: http://www.hebrew4christians.com/Grammar/Unit_Seven/Prepositions_with_Suffixes/prepositions_with_suffixes.html