Federated Learning (FL) is a collaborative machine learning (ML) approach, where multiple clients participate in training an ML model without exposing the private data. Fair and accurate assessment of client contributions is an important problem in FL to facilitate incentive allocation and encouraging diverse clients to participate in a unified model training. Existing methods for assessing client contribution adopts co-operative game-theoretic concepts, such as Shapley values, but under simplified assumptions. In this paper, we propose a history-aware game-theoretic framework, called FLContrib, to assess client contributions when a subset of (potentially non-i.i.d.) clients participate in each epoch of FL training. By exploiting the FL training process and linearity of Shapley value, we develop FLContrib that yields a historical timeline of client contributions as FL training progresses over epochs. Additionally, to assess client contribution under limited computational budget, we propose a scheduling procedure that considers a two-sided fairness criteria to perform expensive Shapley value computation only in a subset of training epochs. In experiments, we demonstrate a controlled trade-off between the correctness and efficiency of client contributions assessed via FLContrib. To demonstrate the benefits of history-aware client contributions, we apply FLContrib to detect dishonest clients conducting data poisoning in FL training.