Leonie is the real first name of the Australian actress Noni Hazlehurst. Madonna's nickname as a child was "Little Nonni" to distinguish her from her mother, who was also named Madonna. I guess Nonni could be pronounced like Nonie.
― Anonymous User 11/11/2013
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Nonie could be a nickname to a plethora of female 'N' names. Natalie, Nona, Nola, Naomi, etc.
It's also diminutive of Natalie. I went to school with a few Latina girls who were named Natalie and went by Nonie. I think it's a really cute nickname.