Hydrodynamic interaction among rigid nanoparticles in laminar flow with Navier's slip boundary condition on the nanoparticles' surfaces is formulated. The single-particle problem under the general linear flow is solved in terms of the Lamb general solution, and the velocity field exerted by the slip particle is expressed in terms of the multipole expansion in the force moments. Thereby, the mobility matrix for a many-body system is constructed with Faxén's laws for the force, torque and the stresslet, and is extended to periodic systems by the Ewald summation technique. Using this formulation, the Stokesian dynamics method is generalized to slip particles with arbitrary slip length. The method is applied to a system in an unbounded fluid and to a system with periodic boundary conditions. The mobility problem with constant force for the former and sedimentation velocity (drag coefficient) and spin and shear viscosities for the latter are solved. A comparison is made with the existing results for no-slip particles. According to the surface slip, the reductions of friction (drag force), spin and shear viscosities are observed for the problems with the applied force, torque, and shear, respectively. In particular, we show that just changing the slip properties of the nanoparticle surface, one can control the drag force within an order of magnitude. The slip-length dependences of the drag coefficient and other rheological properties are useful for rational design of nanofluidic devices, including controllable manipulation and separation of large biomolecules in nanofluidic channels.