Calculating the potential on a surface from the potential on another surface

I think a good phrase to search on would be "Solving the Laplace equation with Dirichlet boundary conditions."

(Though this will give you the potential everywhere, not just on a surface.)

This Wikipedia article discusses existence and uniqueness. (Usually you'll have both in a real physical problem.)

The key to solving this is using the multipole expansion of the charge distribution, at least in the case where $S_1$ is spherical. Since both $S_1$ and $S_2$ are outside the distribution, you can write the potential there as $$ V(r,\theta,\phi) = \sum_{l=0}^\infty \sum_{m=-l}^l Q_{lm} \frac{ Y_{lm}(\theta, \phi) }{r^{l+1}}, $$ with respect to some fixed origin inside the distribution. Here the $Q_{lm}$ are the multipole moments, which can be obtained as integrals over the charge distribution, but you can also get them from the potential itself.

To do this, you just need to do a surface integral along $S_1$ of $V$ against a suitable spherical harmonic, making sure that the surface element reduces strictly to the solid angle on the unit sphere. In particular, take $$ I_{lm} = \int_{S_1} V(\mathbf r)r^{l+1}Y_{lm}(\theta,\phi)^* \mathrm d\Omega, $$ where $\mathrm d\Omega= \displaystyle\frac{\hat{\mathbf r}\cdot\mathrm d\mathbf S}{r^2}$. Substituting in the series and using the orthogonality of the spherical harmonics, you get that $I_{lm}=Q_{lm}$, i.e. you recover full information on $V$ using only its dependence along $S_1$.