Quillen et al. and O'Dea et al. carried out a Spitzer study of a sample of 62 brightest cluster galaxies (BCGs) from the ROSAT brightest cluster sample, which were chosen based on their elevated Hα flux. We present Hubble Space Telescope Advanced Camera for Surveys far-ultraviolet (FUV) images of the Lyα and continuum emission of the luminous emission-line nebulae in seven BCGs found to have an infrared (IR) excess. We confirm that the BCGs are actively forming stars which suggests that the IR excess seen in these BCGs is indeed associated with star formation. Our observations are consistent with a scenario in which gas that cools from the intracluster medium fuels the star formation. The FUV continuum emission extends over a region ~7-28 kpc (largest linear size) and even larger in Lyα. The young stellar population required by the FUV observations would produce a significant fraction of the ionizing photons required to power the emission-line nebulae. Star formation rates estimated from the FUV continuum range from ~3 to ~14 times lower than those estimated from the IR, however, both the Balmer decrements in the central few arcseconds and detection of CO in most of these galaxies imply that there are regions of high extinction that could have absorbed much of the FUV continuum. Analysis of archival Very Large Array observations reveals compact radio sources in all seven BCGs and kpc scale jets in A-1835 and RXJ 2129+00. The four galaxies with archival deep Chandra observations exhibit asymmetric X-ray emission, the peaks of which are offset from the center of the BCG by ~10 kpc on average. A low feedback state for the active galactic nucleus could allow increased condensation of the hot gas into the center of the galaxy and the feeding of star formation.