Do you know anything about it? I couldn't really find something useful on web.
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According to Maley's Higher Order Approximations to Solutions of Transcendental Systems (1960), the earliest occurrence of the inverse quadratic interpolation for finding roots is in Dandelin's Recherches sur la resolution des equations numeriques (1826) published in Nouveaux mémoires de l'Académie royale des sciences et belles-lettres de Bruxelles. Dandelin was a Belgian mathematician now mostly remembered for a clever construction that proves focal properties of conic sections (Dandelin spheres).
The term "inverse quadratic interpolation" is not used by Dandelin. The earliest mention I found is in the MathSciNet review of Praktische Interpolation höherer Ordnung by Rubbert (1948), and Google Ngram shows that its use does not pick up until 1959, perhaps due to the use on computers. Muller adapted the method for one of the first computers, ILLIAC, in 1956, see Method for Solving Algebraic Equations Using an Automatic Computer. It gained its modern prominence after Brent combined it with bisection to speed up Dekker's algorithm for finding zeros of a function without taking derivatives, see Brent, An algorithm with guaranteed convergence for finding a zero of a function (1971).
