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.