With an accent on the e in Limé it would be the past tense act of filing as in filing your nails or metal. It would be pronounced Lee-MAY. It is hard to say without more information on exactly what the owners were going for in the name.
Without an accent on the e in Limé it is Lime as in the fruit or tree. It would be promounced Leem. It means just what you think it should be, The Red Lime.
I am leaning toward the first interpretation since no self respecting French business would make an obvious error in their establishment name. It would be like putting a picture of a lime on your cafe with the word 'banana' on it.
"Ceterum censeo Islam esse delendam."
Garde la Foi, mes amis! Nous nous sommes les sauveurs de la République! Maintenant et Toujours!
(Keep the Faith, my friends! We are the saviors of the Republic! Now and Forever!)
LonePalm, le Républicain du verre cassé (The Broken Glass Republican)
Back to the Pew Poll https://www.thehighroad.org/index.php?threads/mccain-feingold-congress-hoodwinked-by-pew-foundation.130541/
Thanks for your help.
The accent is definitely there. So it’s “filed red”? So strange. Perhaps it’s some idiomatic expression....
I even found reviews for the restaurant online, but they were no help in explaining the name.