What is the unit (dimension) of the 3-dimensional position space wavefunction $\Psi$ of an electron?

The physical interpretation of the wavefunction is that $|\psi(\vec r)|^2dV$ gives the probability of finding the electron in a region of volume $dV$ around the position $\vec r$. Probability is a dimensionless quantity. Hence $|\psi(\vec r)|^2$ must have dimension of inverse volume and $\psi$ has dimension $L^{-3/2}$.

Think about it, its square integrated over a volume (that is multiplied for infinitesimal volumes and summed over all those volumes) is a pure number ("probability of finding the particle in that volume") therefore $(\text{wave function})^2 \cdot (\text{Length})^3 = (\text{adimensional quantity})$

Thus the square of the wavefunction has the dimension of a $(\text{Length})^{-3}$. So the wavefunction is, dimensionally, a $(\text{Length}) ^{-3/2}$

The three-dimensional integral of the norm square of the wave function is a probability, so it should be dimensionless. Therefore $\text{length}^3[\psi]^2=1$, so $[\psi]=\text{length}^{-3/2}$.