We present the surface density of luminous active galactic nuclei (AGNs) associated with a uniformly selected galaxy cluster sample identified in the 8.5 deg2 Bootes field of the NOAO Deep Wide-Field Survey. The clusters are distributed over a large range of redshift (0 < z < 1.5), and we identify AGN using three different selection criteria: mid-IR color, radio luminosity, and X-ray luminosity. Relative to the field, we note a clear overdensity of the number of AGNs within 0.5 Mpc of the cluster centers at z > 0.5. The amplitude of this AGN overdensity increases with redshift. Although there are significant differences between the AGN populations probed by each selection technique, the rise in cluster AGN surface density generally increases more steeply than that of field quasars. In particular, X-ray-selected AGNs are at least 3 times more prevalent in clusters at 1 < z < 1.5 compared to clusters at 0.5 < z < 1. This effect is stronger than can be explained by the evolving median richness of our cluster sample. We thus confirm the existence of a Butcher-Oemler-type effect for AGN in galaxy clusters, with the number of AGNs in clusters increasing with redshift.