Say you have a set of random (x, y) points on a graph. What would the point be that minimizes the distance of this point to each other one?
I imagine you want to use some kind of least squares method, but I only know how to use that in the context of finding lines of best fit. How can I apply that knowledge here?
The point which minimizes the distance to the furthest point in $X$ is usually called the Chebyshev center of $X$. It is the center of the smallest disk which contains $X$.
There are also other options, for instance minimizing the sum of distances to all points of $X$. This is known as the geometric median (or Fermat-Torricelli point if $|X|=3$).
You can find either with quadratic programming or second-order cone programming, so yes, it is analogous to least squares. For example the Chebyshev center can be solved with $$ \begin{array}{ll} \mathrm{minimize} & r \\ \mathrm{s.t.} & r\geq \|y-x\|_2,\ x\in X. \end{array} $$