An all-electron scalar relativistic calculation on the adsorption of hydrogen molecule onto small copper clusters has been performed by using density functional theory with the generalized gradient approximation (GGA) at PW91 level. Our results reveal that after adsorption of H2 molecule, the Cu-Cu interaction is strengthened and the H-H interaction is weakened, the reactivity enhancement of H2 molecule is obvious. The VIPs, HLGs and VEAs of Cu$_n$H2 clusters show an obvious odd-even oscillation. It is suggested that the H2 molecule is more favourable to be adsorbed by the even-numbered small copper clusters. Meanwhile, the odd-even alteration of magnetic moments is also observed and may be served as the material with tunable code capacity of `0’ and `1' by adsorbing hydrogen molecule onto odd or even-numbered small copper clusters. Some discrepancies of dissociative adsorption between our work and previous works are found and may be understood in terms of the electron pairing effect and the scalar relativistic effect.