Budhna is bottom, base, ground, root. It is the bottom of the sea, the foot of a tree, the base of a post. So the name says serpent of the bottom and leaves the reader to decide which bottom. Three readings are in use. The atmospheric one, which takes the deep as the lower reaches of the middle region and makes him a sky serpent, and which is favoured by the company he keeps in the lists. The oceanic one, which takes it literally and puts him at the floor of the waters, in the position that a great many later Indian serpent figures occupy. And a cosmological one, which takes budhna as the base or foundation of things and reads the deep as the bottom of the world rather than of any particular medium.
This is a real disagreement rather than a blank, which is why the atlas records it as argued and not as unknown. The readings are positions, they have been stated, and the evidence for each of them, thin as it is, can be laid out and compared. That is the difference between this entry and the one for his companion Aja Ekapada, where there is no argument to join.
The name outlived the figure. A later Vaishnava text is titled after him, and the association of serpents with the base of things ran on independently into the cosmology of the great serpent on the cosmic waters. In jyotisha he presides over Uttara Bhadrapada, which is directly after the mansion Aja Ekapada presides over. Two systems apart and two thousand years later, the formula that put the two names together is still putting them together, in order.