I love Naomi, probably because of a wonderful girl that I know named Naomi. I'm not sure whether she pronounces it NAY-oh-mee or NYE-oh-mee, though - she answers to both. I'm always inclined to pronounce it NAY-oh-mee; the NYE-oh-mee feels a bit contrived to me, but that's how my boyfriend says it (meaning that perhaps this isn't a name we should consider, lol).
I've never thought of Mina as a nn for Naomi, but I think it works, really - all the letters are there. :) The Naomi I know goes by Omi on occasion. I think that Mimi would be kind of cute for a little Naomi, too.
Some Naomi combos, spanning a wide range of styles since I'm not sure of your naming style:
Naomi Adelaide
Naomi Agatha
Naomi Alice
Naomi Anna
Naomi April
Naomi Asha
Naomi Bernadette
Naomi Calla
Naomi Charlotte
Naomi Clara
Naomi Clementine
Naomi Colette
Naomi Elise
Naomi Ella
Naomi Esmeralda
Naomi Esther
Naomi Faye
Naomi Giselle
Naomi Hazel
Naomi Helena
Naomi Isadora
Naomi Jane
Naomi Jordana
Naomi Josephine
Naomi Karen
Naomi Lauren
Naomi Lorelei
Naomi Lucille
Naomi Lydia
Naomi Lyra
Naomi Mara
Naomi Margaret
Naomi Mirabelle
Naomi Miranda
Naomi Nicole
Naomi Rebecca
Naomi Valentine
Naomi Wilhelmina
Naomi Willow
Naomi Zelda