Functional magnetic resonance imaging (fMRI) research is routinely criticized for being statistically underpowered due to characteristically small sample sizes and much larger sample sizes are being increasingly recommended. Additionally, various sources of artifact inherent in fMRI data can have detrimental impact on effect size estimates and statistical power. Here we show how specific removal of non-BOLD artifacts can improve effect size estimation and statistical power in task-fMRI contexts, with particular application to the social-cognitive domain of mentalizing/theory of mind. Non-BOLD variability identification and removal is achieved in a biophysical and statistically principled manner by combining multi-echo fMRI acquisition and independent components analysis (ME-ICA). Without smoothing, group-level effect size estimates on two different mentalizing tasks were enhanced by ME-ICA at a median rate of 24% in regions canonically associated with mentalizing, while much more substantial boosts (40-149%) were observed in non-canonical cerebellar areas. Effect size boosting occurs via reduction of non-BOLD noise at the subject-level and consequent reductions in between-subject variance at the group-level. Smoothing can attenuate ME-ICA-related effect size improvements in certain circumstances. Power simulations demonstrate that ME-ICA-related effect size enhancements enable much higher-powered studies at traditional sample sizes. Cerebellar effects observed after applying ME-ICA may be unobservable with conventional imaging at traditional sample sizes. Thus, ME-ICA allows for principled design-agnostic non-BOLD artifact removal that can substantially improve effect size estimates and statistical power in task-fMRI contexts. ME-ICA could mitigate some issues regarding statistical power in fMRI studies and enable novel discovery of aspects of brain organization that are currently under-appreciated and not well understood.