We study the problem of joint offloading decision and resource allocation for mobile cloud networks with a computing access point (CAP) and a remote cloud center. We consider the case where the task computing requirement is not fully known before their execution. We aim to jointly optimize the offloading decisions as well as the allocation of computation and communication resources, to minimize a weighted sum of the average cost and cost variation. The problem is formulated as a mixed-integer program. We propose an efficient algorithm, termed Task Offloading and Resource Allocation with Uncertain Computing (TORAUC), and show that it always converges to a Karush-Kuhn-Tucker (KKT) point of an alternate form of the original problem, which has its binary constraints removed but guarantees an offloading decision solution that is arbitrarily close to binary. We extend TORAUC to TORAUC-MP for the case of a multi-processor CAP. Through trace-based simulation, we study the performance of TORAUC and TORAUC-MP. We observe that TORAUC is nearly optimal, and both algorithms substantially outperform several alternatives.