•Transition pressure from B3 to B8 of BeSe is 50.24 GPa.•B3 and B8 phases of BeSe feature an indirect band gap.•Elastic properties of B8 under pressure were predicted for the first time.•Elastic moduli of BeSe increase monotonically with the pressure increasing.•Elastic anisotropy and Debye temperature were investigated.The structural, electronic and elastic properties of BeSe in both B3 and B8 structures have been studied by first-principles calculations within the generalized gradient approximation (GGA). The calculated lattice parameters and bulk modulus of BeSe are in reasonable agreement with previous results. The predicted value of phase transition pressure from B3 to B8 is 50.24 GPa, which is well in line with the experimental data (56 ± 5 GPa). The calculation of the electronic band structure shows that the energy gap is indirect for B3 and B8 phases. Especially, the elastic constants of B8 BeSe under high pressure were studied for the first time. The bulk modulus, shear modulus, compressional and shear wave velocities of B8 BeSe evaluated from elastic constants as a function of pressure were investigated. In addition, Poisson's radio, elastic anisotropy and Debye temperature were analyzed successfully.