The tower carries the name of the man who built it, not the men who dreamed it up.
Wikipedia says the tower is named after the engineer Gustave Eiffel, whose company designed and built it from 1887 to 1889. The official Eiffel Tower site says the idea of a very tall metal tower came from his engineers Maurice Koechlin and Émile Nouguier. Wikipedia reports that Koechlin sketched the idea in May 1884, while the official site gives 6 June 1884. Wikipedia adds that Stephen Sauvestre, head of the company's architecture department, later added decorative arches and a glass pavilion.
Eiffel at first rejected the project but let his team keep developing it, according to the official site. Wikipedia says that once the design was revised, he bought the patent rights that Koechlin, Nouguier and Sauvestre had filed. His name was already on the firm, and it ended up on the tower.
It might not have gone that way. On 14 February 1887 Le Temps published a petition titled "Artists against the Eiffel Tower," denouncing the "useless and monstrous Eiffel Tower." Wikipedia lists Guy de Maupassant and Charles Gounod among the members of the committee behind it. Eiffel replied that his tower would be the tallest structure ever erected by man.
The tower opened to the public on 15 May 1889. It was meant to be temporary, with dismantling planned for 1909. Wikipedia says it was spared because of its value for radio telegraphy.
The name itself has a story. The official site says the family was of German origin and bore the surname Bönickhausen. In the early 19th century family members began using the nickname Eiffel, after the Eifel region they came from. Wikipedia says Gustave was registered at birth as "Bonickhausen dit Eiffel." Both sources give 1880 as the year the name was formalized.
The name has also lasted longer than Eiffel's place on the company. Wikipedia says that after the Panama Canal scandal he resigned from the board in 1893 and asked that his name be removed from the company name. The tower kept it.