We report on the spin dynamics and discovery of magnetoelectricity in the coupled-spin tetrahedral compound Cu2Te2O5Cl2. Te-125 NMR measurements show an anomalous resonance frequency shift and a signal wipe-out phenomenon around the Neel temperature T-N = 18.2 K, which could be attributed to the anomalous critical slowing down of the Cu spin fluctuations on the NMR time scale (similar to 10-100 MHz). The critical exponent of (T1T)(-1) proportional to (T - T-N)(-alpha) is 0.40 +/- 0.03, as compared to 0.5 for a three-dimensional mean-field model. This is in contrast to the Br compound [S.-H. Baek et al., Phys. Rev. B 86, 180405 (2012)], which exhibits pronounced singlet dynamics with a large spin gap. Electric polarization (P-c) is observed along the c axis for temperatures below T-N under finite magnetic field but not sensitive to the electric poling. P-c increases sharply over zero to 2T and then reaches saturation. Below T-N, P-c changes its sign depending on the applied magnetic field direction, positive for the H perpendicular to c axis and negative for H parallel to c axis. We discuss possible explanations for the observed magnetoelectric (ME) behavior in terms of linear ME effect, spin-driven multiferroicity, and an exchange striction of intertetrahedral exchange paths involving the Te4+ lone-pair ions. Our results suggest that Cu2Te2O5Cl2 is a type of ME material whose properties are tuned by intertetrahedral exchange interactions involving polarizable Te4+ ions.