Most people don't associate the letter F with fashionable baby names, because most F names are fashionable precisely because they're UNfashionable. Names such as Florence and Frances, Floyd and Franklin have geek-chic style that appeals to parents searching for an offbeat name for their daughter or son.
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Along with Florence and Franklin, other cool baby names that start with F in the US Top 1000 include Felicity, Finnegan, Flora, Fisher, Frida, Ford, Faye, and Fletcher. Undiscovered F names that are popular internationally include Fraser, Faustine, Floris, and Fiadh. 
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F is a fun, quirky initial with many cool options ranging from familiar to obscure. Some of the <a href="https://nameberry.com/baby-name-lists/category/cool-names">cool baby names</a> beginning with F include those here.

Cool Baby Names That Start With F

A twee German name likely originating as a diminutive of the word frau, meaning "lady, woman, wife".

Frauke

Fidelity is an admirable virtue, yes, but as a name it tends to sound like a financial institution.  Faith, Honor, or Patience would be easier to live with than Fidelity. 

Fidelity

Cool baby names for boys and girls that start with the letter F, including Franklin, Fisher, Flora, and Fable (Page 6)

Cool Baby Names That Start With F

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