easy tumblr answer: People in the Renaissance were wowed by Leonardo's being declared 2nd place in vasari's competition, his 2d optical illusions, superior anatomy, and his new oil techniques looked so realistic. People are interested in Lisa now because of her massive pop culture history and the social capital you can get by visiting her. mostly because of Warhol and other pop art and the big museum heists, she is in every major TV show, movie, cartoon, comic book. Lisa and that one section of the Sistine chapel on the ceiling are the only 2 complete ren paintings most people know about. And you can only take a selfie with one :)
art history 101 answer: leonardo came closest to finishing 3 portraits, Lisa is the most mature surviving example of his unique sfumato oil colore technique which he developed in conversation with the Dutch, Flemish, and German masters. Leonardo's major invenzione, first with his intellectual humanist friend Ginevra, is that he painted women's eyes and head in 3/4 towards the viewer instead of demure and downwards or in a profile box (Botticelli was the worst at the objectifying 'women in frames' topos). (I don't think Leonardo is "protofeminist" because men are not feminists. he was a supporter of women)
Leonardo was sensitive and excited to paint a portrait for a real love marriage, Lisa and Francesco del Giocondo, which was super rare in the Renaissance. he was extremely interested in capturing female interiority and inner psychology, not interested in the sexy blondeness of petrarchan beauties or idealizing women at all, just their realistic faces and the depth behind their eyes (but also he was just not sexually interested in women which is a huge factor). He cleverly added her new surname in the portrait with the smile, Giocondo means to be happy :)
He carried it everywhere and kept experimenting with it as his passion project because Lisa ultimately didn't want it in her house anymore. maybe she wasn't so happily married to the creepy older guy in her neighborhood. :(