The family of the superconducting quasiskutterudites (Ca(x)Sr(1-x))(3)Rh(4)Sn(13) features a structural quantum critical point at x(c)=0.9, around which a dome-shaped variation of the superconducting transition temperature T(c) is found. Using specific heat, we probe the normal and the superconducting states of the entire series straddling the quantum critical point. Our analysis indicates a significant lowering of the effective Debye temperature on approaching x(c), which we interpret as a result of phonon softening accompanying the structural instability. Furthermore, a remarkably large enhancement of 2Δ/k(B)T(c) and ΔC/γT(c) beyond the Bardeen-Cooper-Schrieffer values is found in the vicinity of the structural quantum critical point. The phase diagram of (Ca(x)Sr(1-x))(3)Rh(4)Sn(13) thus provides a model system to study the interplay between structural quantum criticality and strong electron-phonon coupling superconductivity.