There's nothing to do with Hanoi. The French people dislike complicate long names. Professor Lucas came back from Hanoi, Tonkin, where he invented it, so the game was renamed for short "La Tour de Hanoï", thus still keeping some exoticism. Anyway, original namings were made up of anagrams.
"It is perhaps surprising that the origin of the Tower of Hanoi puzzle is still debated...."
Lucas marketed it from 1883 as 'The authentic brain-teaser of the Annamites, a game brought back from Tonkin by Prof. N. Claus of Siam, Mandarin of the College of Li-Sou-Tsian'
QUOTE
"N. CLAUS DE SIAM professeur à LI-SOU-TSIAN", anagramme de "LUCAS D'AMIENS professeur à SAINT-LOUIS".
There are two anagrams in the French: Lucas was born in Amiens, so "N.Claus d'Amiens" is Lucas; meanwhile "Li-Sou-Tsian" is an anagram for Saint Louis.
The exoticisms and in-jokes continue in the instructions, which start
"This game was found, for the first time, in the writings of the illustrious Mandarin FER-FER-TAM-TAM, which are to be published sooner or later, by order of the government of China."
Lucas regarded Fermat as 'one of the greatest geniuses of mankind' and was on the commission for the publication of Fermat's works. From these, it seems unlikely that anyone other than Lucas came up with the puzzle. There is one persistent legend concerning the origin of the Tower of Hanoi.
In "Récréations Mathématiques" (trò vui toán học, Let's Play with Maths), however, Lucas wrote that the legend and the puzzle itself had both been thought up "recently" in Paris, and at the same time that (his nephew) Raoul Olive was the nephew of the inventor
The legend? One version of the story is that under the dome of the great temple at Benares (in India), there are three needles fixed in a brass plate. The Creator placed a sixty-four disk tower on one of these needles at the Creation. Priests transfer disks according to the sacred rules, and when all sixty-four disks have been transferred to another needle, the Tower and Brahmins will fall and the world come to an end.
Thus the puzzle is sometimes known as the Tower of Brahma or the End of the World puzzle. Now a sixty-four disc tower requires M64= 264- 1 > 1019 moves before it is reassembled on another peg. One move each second (and no mistakes) would see the task completed in more than 500 billion years, whatever that means.