We investigate the signature of primordial non-Gaussianities in the weak lensing bispectrum, in particular the signals generated by local, orthogonal and equilateral non-Gaussianities. The questions we address include the signal-to-noise ratio generated in the Euclid weak lensing survey (we find that the 1σ errors for fNL are 200, 575 and 1628 for local, orthogonal and equilateral non-Gaussianities, respectively), mis-estimations of fNL if one chooses the wrong non-Gaussianity model (mis-estimations by up to a factor of ±3 in fNL are possible, depending on the choice of the model), the probability of noticing such a mistake (improbably large values for the χ2-functional occur from fNL∼ 200 onwards), degeneracies of the primordial bispectrum with other cosmological parameters (only the matter density Ωm plays a significant role), and the subtraction of the much larger, structure formation generated bispectrum. If a prior on a standard cold dark matter parameter set including the dark energy equation of state w is available from Euclid and Planck, the structure formation bispectrum can be predicted accurately enough for subtraction, and any residual structure formation bispectrum would influence the estimation of fNL to a minor degree. Configuration space integrations which appear in the evaluation of χ2-functionals and related quantities can be carried out very efficiently with Monte Carlo techniques, which reduce the complexity by a factor of while delivering sub-per cent accuracies. Weak lensing probes smaller scales than the cosmic microwave background (CMB) and hence provides an additional constraint on non-Gaussianities, even though they are not as sensitive to primordial non-Gaussianities as the CMB.