What happens at the exact moment an electron absorbs a photon For example let's have the hydrogen atom in its ground state and the electron gets excited by a photon. If we consider relativity the wave function cannot change instantly, right? The process has to be continous. What happens during this process....
 A: In QM it probably doesn't help to think of an instant - or even of things happening at fixed points in time.
There is an electron if you observe it you will measure a wavefunction. After it absorbs a photon you will measure a different wavefunction.
A: 
For example let's have the hydrogen atom in its ground state and the electron gets excited by a photon. If we consider relativity the wave function cannot change instantly, right?

In general it can, because wave function is just a description of state, and this can change whenever we want, typically when we know better wave function for the job. Relativity does not reject instant changes in descriptions, it only rejects universal simultaneity of events at different places.
But in the case of absorption of light by an atom, if the wave function satisfies  Schroedinger differential equation, it evolves continuously, there is no sudden jump (we do not see what really happens, we just calculate $\psi$, and it evolves continuously.)

The process has to be continous. What happens during this process.... 

It does not have to be continuous. It could be jump-like. We do not see what is really going on at nano-level. But the continuous description is more satisfactory and general (dif. equation vs. probabilistic rule for jump). What happens in continuous description is that the wave function changes to resemble more higher eigenfunctions, particularly those with the Hamiltonian eigennumbers $E$ close to match with the condition
$$
E - E_1 = \hbar \omega
$$ 
where $\omega$ is the angular frequency of the light. Wave function will oscillate in a complicated manner (non-periodically).
