Choose a room in the house without windows. Sound travels through glass too easily to make a room with windows a viable choice. The basement is a good choice because it's more sound-proof and because masonry (i.e. the foundation of the house) works better with the low frequency of the human voice.
Install chipboard over the existing wall. Place a layer of foam or rubber in between the wall and chipboard, then secure the chipboard with screws. Chipboard helps soundproof the room while improving the vocal acoustics.
Drape acoustic foam over the chipboard to further enhance the sound quality in the practice room.
Place thick carpeting on the floor. This is another acoustic and soundproofing technique.
Install a mirror in the room, preferably a full-length mirror. Vocalists should be able to observe themselves singing to improve their aesthetic as well as musical performance.