The connection between the neutrinoless double beta ((\beta\beta)_{0 \nu}-) decay effective Majorana mass, |M_{ee}|, and the branching ratios of the decays H^{\pm\pm}\to l^\pm {l'}^\pm, l,{l'}= e,\mu, of the doubly charged Higgs boson H^{\pm\pm} is analysed within the Higgs Triplet Model of neutrino mass generation. We work in the version of the model with explicit breaking of the total lepton charge conservation, in which H^{\pm\pm}\to l^\pm {l'}^\pm, l,{l'} = e,\mu,\tau, are the dominant decay modes of H^{\pm\pm}. It is assumed also that H^{\pm\pm} are relatively light so that they can be produced at LHC and the branching ratios of interest measured. Taking into account the current and prospective uncertainties in the values of the neutrino mixing parameters most relevant for the problem studied - the atmospheric neutrino mixing angle \theta_{23} and the CHOOZ angle \theta_{13}, and allowing the lightest neutrino mass and the CP violating Dirac and Majorana phases to vary in the intervals [0, 0.3 eV] and [0, 2\pi], respectively, we derive the regions of values of BR(H^{\pm\pm}\to e^\pm e^\pm) and BR(H^{\pm\pm}\to e^\pm \mu^\pm) for which |M_{ee}|\geq 0.05 eV, or |M_{ee}| < 0.05 eV. This is done for neutrino mass spectrum with normal ordering, inverted ordering and in the case when the type of the spectrum is not known, and i) without using the possible additional data on BR(H^{\pm\pm}\to \mu^\pm \mu^\pm), ii) using prospective data on BR(H^{\pm\pm}\to \mu^\pm \mu^\pm). In the latter case results for several values of BR(H^{\pm\pm}\to \mu^\pm\mu^\pm) are presented.