If M is a finite volume complete hyperbolic 3-manifold, the quantity A_1(M) is defined as the infimum of the areas of closed minimal surfaces in M. In this paper we study the continuity property of the functional A_1 with respect to the geometric convergence of hyperbolic manifolds. We prove that it is lower semi-continuous and even continuous if A_1(M) is realized by a minimal surface satisfying some hypotheses. Understanding the interaction between minimal surfaces and short geodesics in M is the main theme of this paper.