I don't believe that there is an immediate connection between waves and qubits.
Most of the terminology is probably due to the historical development of quantum mechanics. For instance, you may have heard about waves when talking about the two slit experiment, where you shoot electrons through a pair of slits and you get an interference pattern on the screen.
In that case, the Schrödinger equation tells you that the electron is described by a propagating wave with a specific frequency $f$ which depends on the electron's energy.
Then, you can define the wave function $\psi(x)$ with amplitude $|\psi(x)|$. The intensity of the wave at some position in space is $|\psi(x)|^2$, which is equal to the probability density of finding the electron at $x$.
$$\rho(x) = |\psi(x)|^2$$
A qubit state is defined as:
$$a|0\rangle + b|1\rangle$$
Here $a$ and $b$ are called amplitudes. This connects to the terminology developed for the double slit experiment in the sense that $|a|^2$ is the probability of finding the qubit in state $0$ and $|b|^2$ of finding the qubit in state 1. Hence the modulus square of the amplitude gives you the probability.
The qubit can represent something like the state of the electron on atom, where $|0\rangle$ corresponds to the electron being in the ground state (close to the nucleum) and $|1\rangle$ to the electron being in an excited state. However, qubits can represent other quantum systems as well, such as photon polarization!