What gives mass to dark matter particles? Assuming that dark matter is not made of WIMPs (weakly interacting massive particles), but interacts only gravitationally, what would be the possible mechanism giving mass to dark matter particles? If they don't interact weakly, they couldn't get mass from interacting with the Higgs field. The energy of gravitational interactions alone does not seem to be sufficient to account for a large particle mass. Would this imply that dark matter consists of a very large number of particles with a very small mass, perhaps much smaller than of neutrinos? Or do we need quantum gravity to explain the origin of mass of dark matter?
 A: There are various ways dark matter could acquire mass that have nothing to do with the standard model weak force. For example, there are theories involving a hidden sector- particles that do not interact with the standard model gauge bosons at all, but have their own interactions.
Note that the Higgs mechanism is not required for all mass generation in the standard model. The massive gauge bosons acquire their mass through the Higgs mechanism, but there are models where the fermionic masses are acquired through different mechanisms. The source of mass for neutrinos in particular is unknown. 
Without knowing what dark matter is, it is of course impossible to determine how it acquires mass.
If it has no interactions at all, there's no need for a mechanism to acquire mass. Explicit mass terms in the standard model Lagrangian are only a problem because they break gauge symmetry. If a field doesn't couple to the gauge fields, its mass terms don't break gauge symmetry, and the mass can just be added to the Lagrangian by hand.
A: I think this question contains a misconception unfortunately caused by popular science descriptions of the Standard Model. 
The question seems to assume there needs to be some concrete source that particles "get" mass from, as if mass is a resource like money and the Higgs field is giving it out. But that's not right. In a generic field theory there is no issue adding a new field $\psi$ whose particles have mass. The only thing you have to do is make sure the Lagrangian has a term proportional to $\psi^2$.
You might protest that this violates the conservation of energy because the mass has to "come from" somewhere, but that's not right. Mass is the energy price for creating a particle. I don't create money by changing the pricetag of an item in a store. 
The reason science popularizers say that mass must come from the Higgs mechanism is because of a peculiarity of the Standard Model (SM). The symmetries of the SM forbid a term such as $\psi^2$ for any field $\psi$ in the SM, so we need a trick to get a mass term. In brief, the Higgs field $\phi$ allows us to write terms like $\phi \psi^2$ which do respect the symmetry. This is an interaction term, but we can set up the Lagrangian so the Higgs field $\phi$ acquires a constant part, yielding the $\psi^2$ mass term we wanted.
However, once you start speculating about dark matter models, especially dark matter that does not interact with the electroweak force at all, these constraints don't apply and generically there is nothing forbidding a $\psi^2$ term. There's no need for any special mechanism for "giving" mass. You just treat mass exactly like you did in high school, intro mechanics and quantum mechanics: write it down, call it $m$ and call it a day. 
