In this paper, we consider the weighted online set k-multicover problem. In this problem, we have a universe V of elements, a family S of subsets of V with a positive real cost for every set in S and a "coverage factor" (positive integer) k. A subset of elements are presented online in an arbitrary order. When each element, say i, is presented, we are also told the collection of all (at least k) sets and their costs to which i belongs and we need to select additional sets from these sets containing i, if necessary, such that our collection of selected sets contains at least k sets that contain the element i. The goal is to minimize the total cost of the selected sets (our algorithm and competitive ratio bounds can be extended to the case when a set can be selected at most a pre-specified number of times instead of just once; we do not report these extensions for simplicity and also because they have no relevance to the biological applications that motivated our work). In this paper, we describe a new randomized algorithm for the online multicover problem based on a randomized version of the winnowing approach of Littlestone. This algorithm generalizes and improves some earlier results by N. Alon, B. Awerbuch, Y. Azar, N. Buchbinder, and J. Naor. We also discuss lower bounds on competitive ratios for deterministic algorithms for general $k$.