The Collaborative International Dictionary of English v.0.48:

Nanny \Nan"ny\, n.
   1. A diminutive of Ann or Anne, the proper name.
      [1913 Webster]

   Nanny goat, a female goat. [Colloq.]
      [1913 Webster] Nanny

The Collaborative International Dictionary of English v.0.48:

Nanny \Nan"ny\, Nannie \Nan"nie\(n[a^]n"n[y^]), n.; pl.
   nannies.
   1. a caretaker for a child; a child's nurse; a nursemaid.
      [PJC]

   2. Grandmother; -- a child's word, used especially as a form
      of address. See also nana.
      [PJC]

WordNet (r) 3.0 (2006):

nanny
    n 1: a woman who is the custodian of children [syn: nanny,
         nursemaid, nurse]
    2: female goat [syn: nanny, nanny-goat, she-goat]