In this paper, the motion of a circular particle in a lid-driven square cavity with the power-law fluids is studied by the diffuse interface lattice Boltzmann method, and the effects of the initial position of particle, power-law index, Reynolds number and particle size are mainly considered. The numerical results show that the circular particle is first in a centrifugal motion under the effect of inertia, but finally, it moves steadily on the limit cycle. Furthermore, it is also found that the initial position of the particle has no influence on the limit cycle.For shear-thinning fluid flows, the limit cycle moves towards the bottom right corner of the square cavity. Moreover, the particle velocity is small, and the period of the particle motion is long. On the other hand, for shear-thickening fluid flows, the limit cycle moves towards the top left corner of the cavity. In addition, the particle velocity is large, and the period of the particle motion is short.With the increase of Reynolds number, the limit cycle moves towards the bottom right corner of the square cavity, which is caused by a strong fluid flow field. Meanwhile, the particle velocity becomes larger, and the period of the particle motion is shorter. With the increase of particle size, the effect of confinement of the cavity boundary becomes significant, and the circular particle is pushed toward the center of the cavity. In this case the limit cycle shrinks toward the center of the cavity. The circular particle squeezes the secondary vortices, especially when the circular particle is located in the left bottom, right bottom and left upper corners. Additionally, The appearance of the circular particle has a significant influence on the position of the primary vortex, which changes periodically around the position of the primary vortex without the particle. It is also found that the effect of the circular particle is more significant with the increase of the particle size and the decrease of the power-law index.