The thermostatistic properties of a q-generalized boson system trapped in an n-dimensional harmonic oscillator potential are studied, based on the generalized statistic distribution derived from Tsallis' entropy. The density of states, total number of particles, critical temperature at which Bose-Einstein condensation occurs, internal energy, and heat capacity at constant volume are derived. The characteristics of Bose-Einstein condensation of the system are discussed in detail. It is found from the results obtained here that the thermostatistic properties of such a system depend closely on parameter q, dimensional number of the space, and frequency of the harmonic oscillator; and the external potential has a great influence on the thermostatistic properties of the system.