Identifying the source of network diffusion is an important task in applications such as epidemics management and understanding the trend propagation over social networks. As observing each node carries a cost, we study the problem of sequential selection of observed nodes from two aspects: which nodes to observe such that the source is localized with the lowest cost, and for a pre-specified number of time-steps, which nodes to observe such that the resulting number of possible source candidates is the lowest. We show that both problems can be framed, under a simple propagation scenario, as dynamic programing with imperfect state knowledge. The proposed approach is optimal, but computationally intensive, hence we propose two simple greedy strategies. Using adaptive submodularity, we provide performance guarantees for one greedy algorithm. We evaluate the proposed approaches through simulation.