We present the results of an analysis of Wide-field Infrared Survey Explorer (WISE) observations of the full 2500 deg2 South Pole Telescope (SPT)-Sunyaev-Zel'dovich cluster sample. We describe a process for identifying active galactic nuclei (AGN) in brightest cluster galaxies (BCGs) based on WISE mid-IR color and redshift. Applying this technique to the BCGs of the SPT-SZ sample, we calculate the AGN-hosting BCG fraction, which is defined as the fraction of BCGs hosting bright central AGNs over all possible BCGs. Assuming an evolving single-burst stellar population model, we find statistically significant evidence (>99.9%) for a mid-IR excess at high redshift compared to low redshift, suggesting that the fraction of AGN-hosting BCGs increases with redshift over the range of 0 < z < 1.3. The best-fit redshift trend of the AGN-hosting BCG fraction has the form (1 + z)4.1±1.0. These results are consistent with previous studies in galaxy clusters as well as as in field galaxies. One way to explain this result is that member galaxies at high redshift tend to have more cold gas. While BCGs in nearby galaxy clusters grow mostly by dry mergers with cluster members, leading to no increase in AGN activity, BCGs at high redshift could primarily merge with gas-rich satellites, providing fuel for feeding AGNs. If this observed increase in AGN activity is linked to gas-rich mergers rather than ICM cooling, we would expect to see an increase in scatter in the P cav versus L cool relation at z > 1. Last, this work confirms that the runaway cooling phase, as predicted by the classical cooling-flow model, in the Phoenix cluster is extremely rare and most BCGs have low (relative to Eddington) black hole accretion rates.