Triply periodic minimal surfaces (TPMS) provide excellent stiffness-to-weight characteristics and are well suited to additive manufacturing, yet their geometric periodicity often yields elastic anisotropy that can undermine performance under uncertain or multiaxial loading. Here we propose TPMS-based aperiodicity-induced architected materials (TAAMs), which embed controllable geometric disorder ("designable disorder") as an explicit design variable to simultaneously improve stiffness and elastic isotropy. Starting from Schoen’s I-Wrapped Package (IWP) surface, TAAM unit cells are generated by combining (i) anisotropic shape modulation via scaling factors (α, β, γ) and (ii) randomized unit-cell rotations (𝜃x, 𝜃y, 𝜃z) within prescribed bounds, thereby expanding the design space while preserving manufacturability. For each candidate, the effective stiffness tensor is evaluated by finite- element computational homogenization under six independent affine load cases, and isotropy is quantified by ζ = Emin/Emax computed from directional Young’s moduli. To efficiently navigate the stiffness–isotropy trade-off across relative densities ρ = 0.2–0.4, we couple a Gaussian-process surrogate model (Matérn 5/2 kernel) with multi-objective Bayesian optimization, using expected and probability hypervolume improvement to robustly identify Pareto-optimal designs. The optimized TAAMs consistently dominate periodic TPMS baselines and prior isotropic TPMS designs, approaching the Hashin–Shtrikman upper bound for isotropic composites; for ρ = 0.3, ζ increases from ~0.86 to >0.94 with only a marginal change in average stiffness. Selected initial and optimized TAAMs were fabricated by fused filament fabrication (PLA) and validated by uniaxial compression along three principal axes, showing strong agreement with numerical predictions and markedly reduced directional variability in modulus and collapse metrics. These results demonstrate designable disorder as a scalable, data-driven route to stiff yet nearly isotropic architected materials.