  We explore the effect of noise on the ballistic graphene-based small
Josephson junctions in the framework of the resistively and capacitively
shunted model. We use the non-sinusoidal current-phase relation specific for
graphene layers partially covered by superconducting electrodes. The noise
induced escapes from the metastable states, when the external bias current is
ramped, give the switching current distribution, i.e. the probability
distribution of the passages to finite voltage from the superconducting state
as a function of the bias current, that is the information more promptly
available in the experiments. We consider a noise source that is a mixture of
two different types of processes: a Gaussian contribution to simulate an
uncorrelated ordinary thermal bath, and non-Gaussian, $\alpha$-stable (or
L\'evy) term, generally associated to non-equilibrium transport phenomena. We
find that the analysis of the switching current distribution makes it possible
to efficiently detect a non-Gaussian noise component in a Gaussian background.
