Assessment of source‐specific health effects has received growing attention in air pollution epidemiology over the past decade. Regardless of inherent uncertainty in the assessment of source‐specific exposures, only a handful of previous studies coped with model uncertainty in source apportionment and/or accounted for exposure measurement error in the estimation of health effects, all under normal health outcome models. We present a source‐specific health effects evaluation approach within a Bayesian framework that can handle both parameter uncertainty and model uncertainty in source apportionment under Poisson health outcome models for low daily mortality count data. While the use of a Poisson health outcome model is apparently more appropriate for low daily mortality count data for which normal approximation is not justified, it introduces additional complexity in estimating model uncertainty. We handle this complexity by introducing appropriate latent variables. The proposed method is illustrated with simulated data and daily ambient concentrations of the chemical composition of fine particulate matter (PM2.5), weather data, and counts of deaths from pneumonia in older adults (≥65 years of age) in Houston, Texas, from January 2002 to August 2005.