Let \(C\) be a reduced complete algebraic curve with at most ordinary double points as singularities, defined over the field of complex numbers. In the present brief note, the authors consider double coverings of \(C\), which are non-trivial on each irreducible component of \(C\) and satisfy the so-called Beauville condition with respect to the singular loci [cf. A. Beauville, Invent. Math. 41, 149-196 (1977; Zbl 0333.14013)]. In this case, the (generalized) Prym variety of such a double cover (which is then called a “Beauville pair”) is well-defined as a principally polarized abelian variety, and a natural problem is to classify all Prym varieties that arise from such specified double covers. Following the approach of V. V. Shokurov [cf. Math. USSR, Izv. 23, 93-147 (1984); translation from Izv. Akad. Nauk SSSR, Ser. Mat., 47, No. 4, 785-855 (1983; Zbl 0572.14025)], the authors investigate the particular case of double covers of hyperelliptic curves, and obtain a complete characterization of all Prym varieties associated with them. A partial result in this direction had been obtained by S. Pantazis [cf. Math. Ann. 273, 297-315 (1986; Zbl 0566.58028)], and the author’s results cover now the remaining cases. At the end of the paper, an application is given to construct finite-zone solutions of the (nonlinear) Landau-Lifshitz equation in terms of the Prym theta functions associated with double covers of elliptic curves.