Strong gravitational lensing is a powerful technique for probing galaxy mass distributions and for measuring cosmological parameters. We present a pixelated approach to modeling simultaneously the lens potential and source intensity of strong gravitational lens systems with extended source-intensity distributions. For systems with sources of sufficient extent such that the separate lensed images are connected by intensity measurements, the accuracy in the reconstructed potential is solely limited by the quality of the data. We apply this potential reconstruction technique to deep HST observations of B1608+656, a four-image gravitational lens system formed by a pair of interacting lens galaxies. We present a comprehensive Bayesian analysis of the system that takes into account the extended source-intensity distribution, dust extinction, and the interacting lens galaxies. Our approach allows us to compare various models of the components of the lens system, which include the point-spread function (PSF), dust, lens galaxy light, source-intensity distribution, and lens potential. Using optimal combinations of the PSF, dust, and lens galaxy light models, we successfully reconstruct both the lens potential and the extended source-intensity distribution of B1608+656. The resulting reconstruction can be used as the basis of a measurement of the Hubble constant. We use our reconstruction of the gravitational potential to study the relative distribution of mass and light in the lensing galaxies. We find that the mass-to-light ratio for the primary lens galaxy is (2.0+/-0.2)h M_{sun} L_{B,sun}^{-1} within the Einstein radius 3.9 h^{-1} kpc, in agreement with what is found for noninteracting lens galaxies at the same scales. (Abridged)
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