Estimating the functional interactions between brain regions and mapping those connections to corresponding inter-individual differences in cognitive, behavioral and psychiatric domains are central pursuits for understanding the human connectome. The number and complexity of functional interactions within the connectome and the large amounts of data required to study them position functional connectivity research as a "big data" problem. Maximizing the degree to which knowledge about human brain function can be extracted from the connectome will require developing a new generation of neuroimaging analysis algorithms and tools. This review describes several outstanding problems in brain functional connectomics with the goal of engaging researchers from a broad spectrum of data sciences to help solve these problems. Additionally it provides information about open science resources consisting of raw and preprocessed data to help interested researchers get started.