The goal of influence maximization (IM) is to select a small set of seed nodes which maximizes the spread of influence on a network. In this work, we propose BOPIM, a Bayesian Optimization (BO) algorithm for IM on temporal networks. The IM task is well-suited for a BO solution due to its expensive and complicated objective function. There are at least two key challenges, however, that must be overcome, primarily due to the inputs coming from a cardinality-constrained, non-Euclidean, combinatorial space. The first is constructing the kernel function for the Gaussian Process regression. We propose two kernels, one based on the Hamming distance between seed sets and the other leveraging the Jaccard coefficient between node's neighbors. The second challenge is the acquisition function. For this, we use the Expected Improvement function, suitably adjusting for noise in the observations, and optimize it using a greedy algorithm to account for the cardinality constraint. In numerical experiments on real-world networks, we prove that BOPIM outperforms competing methods and yields comparable influence spreads to a gold-standard greedy algorithm while being as much as ten times faster. In addition, we find that the Hamming kernel performs favorably compared to the Jaccard kernel in nearly all settings, a somewhat surprising result as the former does not explicitly account for the graph structure. Finally, we demonstrate two ways that the proposed method can quantify uncertainty in optimal seed sets. To our knowledge, this is the first attempt to look at uncertainty in the seed sets for IM.