All I think of is-- Mi-mi-mi-mi-miiii... a singing telegram, I'll sing it really well!Anyway, I only think of the song "A Singing Telegram" from The Backyardigans, a now-ended cartoon that began in 2004.
Mimi is the name of Panda's owner on the anime Hamtaro. She is the little cousin of Laura's best friend Kana. Unlike the other human characters, Mimi is able to communicate with the Ham-Hams.
A masculine use of the name is for the male protagonist of the film The Seduction of Mimi, the original Italian title of which is Mimi metallurgico ferito nell'onore ("Mimi the metalworker wounded in honor.") The character's real name is Carmelo Mardocheo. He is played by Giancarlo Giannini.
Melody Damayo is a model, film director, and actress of Filipino descent best known as Mimi Miyagi. In 2011, Complex magazine ranked her at #12 in their list of “The Top 50 Hottest Asian Porn Stars of All Time”.
Madeleine "Mimi" Force, from the Blue Bloods series by Melissa de la Cruz. I agree, you could use this nickname for almost any M name. My aunt's name is Miriam, and she goes by Mimi.