Brightest cluster galaxies (BCGs) are usually quiescent, but many exhibit star formation. Here we exploit the opportunity provided by rest-frame UV imaging of galaxy clusters in the CLASH (Cluster Lensing and Supernovae with Hubble) Multi-Cycle Treasury Project to reveal the diversity of UV morphologies in BCGs and to compare them with recent simulations of the cool, star-forming gas structures produced by precipitation-driven feedback. All of the CLASH BCGs are detected in the rest-frame UV (280 nm), regardless of their star-formation activity, because evolved stellar populations produce a modest amount of UV light that traces the relatively smooth, symmetric, and centrally peaked stellar distribution seen in the near infrared. Ultraviolet morphologies among the BCGs with strong UV excesses exhibit distinctive knots, multiple elongated clumps, and extended filaments of emission that distinctly differ from the smooth profiles of the UV-quiet BCGs. These structures, which are similar to those seen in the few star-forming BCGs observed in the UV at low redshift, are suggestive of bi-polar streams of clumpy star formation, but not of spiral arms or large, kpc-scale disks. Based on the number of streams and lack of culprit companion galaxies, these streams are unlikely to have arisen from multiple collisions with gas-rich galaxies. These star-forming UV structures are morphologically similar to the cold-gas structures produced in simulations of precipitation-driven AGN feedback in which jets uplift low-entropy gas to greater altitudes, causing it to condense. Unobscured star-formation rates estimated from CLASH UV images using the Kennicutt relation range up to 80 solar masses per year in the most extended and highly structured systems. The circumgalactic gas-entropy threshold for star formation in CLASH BCGs at z~0.2-0.5 is indistinguishable from that for clusters at z