Why don't we call Qbits just "qits". Since Quantum-Binary-Digit doesn't really make sense. Do people in the field refer to them in other ways?
There are also qutrits, which are quantum trits (superpositions of |0>, |1>, and |2>), "qubytes" (collections of 8 qubits), and so on.
This terminology is now 30 years old, and in probably thousands of books and tens of thousands of papers. It's not going to change.
The earlier name for "qubit"---the name that Feynman, for example, would have recognized---was "spin-1/2 particle." But while there was some resistance, I think that even within particle physics, condensed-matter physics, etc., they'd now typically say qubit rather than spin-1/2.
In any case, would you pronounce "qit" like "kit" or "kwit"? :-)
In a qubit the infinitely many superposition states are all mixtures of just two eigenstates.
Further, (and anyone, please correct me where I'm wrong), the eigenfunctions (which could actually be called eigenstates) of an operator ARE the basis set, as they are orthonormal. (Right?)