When Ludwig Nohl discovered "Für Elise" he included the work in his 1867 publication "New Beethoven Letters" (Neue Briefe Beethovens) (pages 28-33). "Für Elise" merely refers to the letter written to Elise, and is a name applied after Nohl, rather than by Nohl himself. For comparison, other collected letters of Beethoven are entitled with the name of the intended recipient.
The French title "Lettre à Elise" ("Letter for Elise", "Brief für Elise") is most likely an association with Nohl's publication owing to the composition being included within the letter.
I have been unable to locate a French translation of "Neue Briefe Beethovens" to verify whether the composition was given this title from a translation of Nohl. The balance of probability is that the title "Lettre à Elise" was applied through common usage and became fixed.
It is unlikely that we will ever know who first gave the work this French title.