Ernő Rubik did not name the Rubik's Cube. A toy company did.
Rubik built his first working model in 1974. Xpat Loop dates it to May 19, and History Daily calls May 19 the date of his second prototype, so the exact day is not settled. He taught architecture in Budapest. The cube is widely said to have been a teaching tool for three-dimensional shapes, but Wikipedia says his real aim was an engineering puzzle about a stable, twistable structure. He called it the Magic Cube, in Hungarian bűvös kocka.
Rubik applied for a Hungarian patent for the Magic Cube on January 30, 1975. The first test batches reached Budapest toy shops in late 1977. In the United Kingdom, a company called Pentangle Puzzles sold it in 1978.
The cube reached the wider world through a trade fair. In February 1979, a Hungarian businessman named Tibor Laczi took one to the Nuremberg Toy Fair. Tom Kremer of Seven Towns noticed it. That September, Seven Towns and Rubik's Hungarian partners signed a deal with the Ideal Toy Company to release it worldwide.
Ideal needed a name that would sell. According to Wikipedia, it considered "The Gordian Knot" and "Inca Gold." In the end, in 1980, the puzzle was renamed Rubik's Cube, after its inventor. Wikipedia says the first officially branded cubes were exported from Hungary in May 1980. The Saturday Evening Post says the renamed toy reached American stores that May.
The sources I read do not say which person at Ideal or Seven Towns made the final choice. They only say the renaming came with the Ideal deal.
The change worked. The Cube won a special Spiel des Jahres award for best puzzle in 1980. Estimates of total sales run to about half a billion cubes, though the figure is approximate.