I will reduce the detailed history to principal highlights. Any of the physicists mentioned below can be considered "fathers" of string theory depending on which aspect of it is considered "the core". The problem with a straightforward answer is that the mathematical formalism was originally developed for purposes that had little to do with modern string theory, and there were several additional insights by different people that made it viable, any of which can be seen as its "true birth".
The earliest candidate for a "father" is perhaps Gabriele Veneziano, who used Euler's beta function as a scattering amplitude in his 1968 paper. At the time unification of quantum fields and gravity was not on the agenda, the issue was working out a theory of strong interaction, and that is what Veneziano introduced his formula for. By 1970, Nambu, Nielsen and Susskind concluded that Veneziano's formula would be based on more than a mere analogy if particles were treated as vibrating strings. This led to the so-called dual resonance models of strong interaction, but they were abandoned in 1973 in favor of quantum chromodynamics.
The original string theory was purely bosonic, fermions were added in 1971 by Ramond and led to the discovery of supersymmetry and superstrings. The number of required dimensions was reduced from 26 in bosonic theory to 10. One of the bosonic vibration modes was a massless particle of spin two, which as Schwarz, Scherk, and independently Yoneya, pointed out in 1974, were exactly the properties of hypothesized graviton. Moreover, in late 1970s supersymmetry was used to unify field theories with gravity independently of strings, in the so-called supergravity. So while original string theory of strong interactions was discarded, the mathematical formalism lived on as a candidate for quantum gravity.
It was not taken seriously however until 1984, when Green and Schwarz discovered that superstring theories produce counterterms that eliminate so-called quantum anomalies. These are mathematical complications that prevent quantized field theories from manifesting classical symmetries, and make them inconsistent. This came to be called the "first superstring revolution", which attracted many new researchers to the field, including Witten, who initiated the "second superstring revolution" in 1995.
