Is it possible to be at rest relative to the cosmic background radiation? Could there be a body in the universe which is at rest relative to the cosmic background radiation? And how much slower would time elapse at this point relative to Earth?
The Milky Way is moving relative to the CMB at approx 2 million km per hour in the direction of the constellation Hydra. If that speed dropped to a very low number then I would consider that 'at rest' or very close to it.
I ask this question because I'd like to know if there are places in the universe with very low relative time dilations to us. It could be very efficient to locate computing power there.
 A: The frame in which the CMB is isotropic is just the rest frame of all the matter in the universe when we average out the peculiar velocities. The CMB is the black body radiation released by the matter (mostly ionised hydrogen) at the time of recombination about 378,000 years after the Big Bang. So its distribution reflects the distribution of the matter at that time. The point is that the frame is defined by the averaged out motion of the matter, and the CMB serves as a convenient label for it - the frame isn't as it is because of the radiation, the radiation is as it is because of the matter.
So yes of course there will be some objects that have a peculiar velocity of zero and therefore are at rest in the frame shared by the CMB. We call these objects comoving. Most objects are not comoving because they have been accelerated by concentrations of mass in their vicinity, but assuming a random distribution of velocities there will be some objects that are comoving (to whatever precision you require).
As you say in a comment the Solar System happens not to be a comoving object, but in relativistic terms its peculiar velocity is not great. According to NASA the peculiar velocity of the Solar System is about 360 km/sec, which is about $0.0012c$ and corresponds to a Lorentz factor of about $1.0000007$. So our relative time dilation compared to an object with a peculiar velocity of zero is only a factor of $1.0000007$.
