Wave function determines complex amplitudes to possible measurement outcomes. The Born Rule states that the probability of obtaining some measurement outcome is equal to the square of the corresponding amplitude.
How did Born arrive at this revolutionary idea? Was he motivitaed by some mathematical principle? Or was it based on pure experimental evidence?
One motivation comes from looking at light waves and polarisation -- when light passes through some filter, the energy of a light wave is scaled by $\cos^2\theta$ -- for a single photon, this means (as you can't have $\cos^2\theta$ of a photon) there is a probability of $\cos^2\theta$ that the number of photons passing through is "1". This $\cos\theta$ is simply the dot product of the "state vector" (polarisation vector) and the eigenvector of the number operator associated with polarisation filter with eigenvalue 1 -- i.e. the probability of observing "1" is $|\langle\psi|1\rangle|^2$, and the probability of observing "0" is $|\langle\psi|0\rangle|^2$, which is Born's rule.
So if you're motivating the state vector based on the polarisation vector, you can motivate Born's rule from $E=|A|^2$, as above.
More abstractly, if you accept the other axioms of quantum mechanics, Born's rule is sort of the "only way" to encode probabilities, as you want probability of the union of disjoint events to be additive (equivalent to the Pythagoras theorem) and the total probability to be one (the length of the state vector is one).
But there is no way to "derive" the Born rule, it is an axiom. Quantum mechanics is fundamentally quite different to e.g. relativity, in the sense that it develops a whole new abstract mathematical theory to connect to the real world. So unlike in relativity, you don't have two axioms that are literally the result of observation and everything is derived from it -- instead, you have an axiomatisation of the mathematical theory, and then a way to connect the theory with observation, which is what Born's rule is. Certainly the motivation for quantum mechanics comes from wave-particle duality, but this is not an axiomatisation.
Here is an answer to the question posed in the title:
Zurek gave a derivation of Born rule using envariance (invariance due to entanglement).
Also, Saunders gave a derivation of Born rule from operational assumptions.
However, the consensus seems to be that there is no generally accepted derivation of the Born rule.
Update: Born rule, has been derived from simpler physical principles in this paper,The Measurement Postulates of Quantum Mechanics are Redundant. The authors state that " Our result shows that not only is the Born rule a good guess, but it is the only logically consistent guess".
Born calculated solution to Schroedinger's equation corresponding to electron scattering experiment and what he got was continuous function of scattering angles measured with respect to the original direction of propagation of electrons.
However, in experiment electrons are always detected at definite points of a screen. Clearly, there is no direct match between the $\psi$ function (continuous) and location of detected electrons (discrete). Born realized one way to resolve this mismatch and make use of the calculated function $\psi$ anyway is to assume it gives continuous probability density for angles the electron goes into, or, more generally, probability density for any possible configuration of particles.
If you want to find a probablity in statistics you have to do the same thing but as for wave function no one was able to interpret it, Born suggested that the same thing is also applied to Quantum wavefunction. Before Born, physicists applied the same thing for photons as well.
Like Schrodinger, he wasn't able to visualize the wave function but he still showed his equation for real waves (such as string or any other mechanical waves), and it fitted quantum waves as well.
The Bohr Model is an excellent model for hydrogen in terms of predicting its spectral lines with classical physics, using a ring-source of charge rotating at a specific frequency and radius from the central atom that minimizes the total energy of the configuration, with the boundary condition of coherent constructive interference of the electron wave going once around the ring.
Born's relation is directly related to physical interpretations, and so he squared the wave function. It is like how in sine waves it doesn't show the physical structures, but $\sin^2$ gives the square of the wavefunction of the electrons shot at a screen.
