Classical Morita equivalence is given by tensoring with a projective bimodule. The obvious generalization to dg algebras is to tensor with a dg bimodule that is some sense projective (the technical term is 'perfect'). But since complexes of modules over a classical algebra are the same thing as dg modules over that classical algebra thought of as a dg algebra with zero differential and concentrated in degree zero, one is led to the notion of derived Morita invariance by thinking of your classical algebra as a very simple kind of dg algebra.
Derived Morita equivalence is thus a generalization of classical Morita equivalence for algebras and enjoys many of the properties of the classical version. For example, every linear invariant that I know of (algebraic K-theory, cyclic (co)homology, Hochschild (co)homology,...) is not only invariant under Morita equivalence but also under derived Morita equivalence. Such invariance is well-known and can be found for example in papers of Bernhard Keller. So I would say that already for classical algebras (with zero differential and concentrated in degree zero), derived Morita invariance is a natural notion, and for more complicated dg algebras it is hard to imagine any other reasonable notion.
I can also very much recommend Stefan Schwede's Morita theory in abelian, derived and stable model categories, in Structured Ring Spectra, 33--86, London Mathematical Society Lecture Notes 315. You can also find this on Schwede's website.