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.
 
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. 
 
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

Friendly, cheerful Irish name that originated as a nickname for a redhead. Potential problem: reminiscent of the Spanish custard. 

Flann

Long before the vogue of using Irish surnames for girls, writer Flannery O'Connor gave this one some visibility. It has a warm (flannelly) feel and the currently popular three-syllable ee-ending sound. 
Mary Flannery O'Connor was an important twentieth century regional writer whose work has been characterized as Southern Gothic, often featuring grotesque characters and concerned with questions of morality.

Flannery

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