Find point with smallest average distance to a set of given points
This point is known as the geometric median, also known as the 1-median.

Apparently, there is no simple algorithm for computing the geometric median.  Instead, one must use some kind of numerical approximation.  Wikipedia mentions Weiszfeld's algorithm, which appears to be a kind of iterative descent algorithm, and cites other more sophisticated algorithms.

This is a special case of the Weber problem, a facility location problem.  Standard solvers for facilitation location problems be able to compute the geometric median for you, by framing this as an instance of the Weber problem.