Potential term in the hydrogen atom Hamiltonian
What is the $r$ there?


$r$ is the same in the potential energy term of the Hamiltonian as in the kinetic energy term of the Hamiltonian.

$r$ is the distance from electron to the proton.


  But isn't the position of the electron non-deterministic?


In the generally accepted Copenhagen interpretation of quantum mechanics, the position of the electron is not deterministic. However, expressing the kinetic and potential energy of the the electron at each position in space does not contradict this interpretation. It is only saying, for each point, if the electron is at this point, the energy is _.   


  Shouldn't that thing be a probability distribution of the distance instead?


The wave function solutions represent probability amplitude in the Copenhagen interpretation.  The square of the absolute value of the wavefunction represents a probability distribution.


  Why isn't it squared?


Force would be inversely proportion to $r^2$, but potential is inversely proportional to $r$.