Goldenrod City, in the north in a tent-like house.
But remember he won't let you rename a Pokemon you obtained in a trade.
Chat with our AI personalities
I think the Name Rater is in Lavender Town.
There is no basement in the radio tower. If you are referring to the underground tunnels it is located near the Name Rater's hut.
You Can Go By The Pokemon Name Rater
Unfortunately there is no way to change the name of your human character once you have chosen it. To change the name of your Pokemon, the name rater should be in Goldenrod next to the Pokemart.
The name rater is in the building on the left in the northern part of Castelia City.