What gives mass to dark matter particles?

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.

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.

From: "A White Paper on keV Sterile Neutrino Dark Matter" (9 Feb 2017), by over 130 authors:

"Executive Summary

Despite decades of searching, the nature and origin of Dark Matter (DM) remains one of the biggest mysteries in modern physics. Astrophysical observations over a vast range of physical scales and epochs clearly show that the movement of celestial bodies, the gravitational distortion of light and the formation of structures in the Universe cannot be explained by the known laws of gravity and observed matter distribution $^{[1–7]}$.

They can, however, be brought into very good agreement if one postulates the presence of large amounts of non-luminous DM in and between the galaxies, a substance which is much more abundant in the Universe than ordinary matter $^{[1]}$. Generic ideas for what could be behind DM, such as Massive Compact Halo Objects (MACHOs) $^{[8–11]}$ are largely ruled out $^{[12, 13]}$ or at least disfavored $^{[14, 15]}$. Alternative explanations based on a modification of the law of gravity $^{[16]}$ have not been able to match the observations on various different scales. Thus, the existence of one or several new elementary particles appears to be the most attractive explanation.

As a first step, the suitability of known particles within the well-tested Standard Model (SM) has been examined. Indeed, the neutral, weakly interacting, massive neutrino could in principle be a DM candidate. However, neutrinos are so light that even with the upper limit for their mass $^{[17, 18]}$ they could not make up all of the DM energy density $^{[19]}$. Moreover, neutrinos are produced with such large (relativistic) velocities that they would act as hot DM (HDM), preventing the formation of structures such as galaxies or galaxy clusters $^{[20]}$.

Consequently, explaining DM in terms of a new elementary particle clearly requires physics beyond the SM. There are multiple suggested extensions to the SM, providing a variety of suitable DM candidates, but to date there is no clear evidence telling us which of these is correct."

...

References:

1. Planck collaboration, P. A. R. Ade et al., Planck 2015 results. XIII. Cosmological parameters, Astron. Astrophys. 594 (2016) A13, [1502.01589].
2. M. Persic, P. Salucci and F. Stel, The Universal rotation curve of spiral galaxies: 1. The Dark matter connection, Mon. Not. Roy. Astron. Soc. 281 (1996) 27, [astro-ph/9506004].
3. S. M. Faber and R. E. Jackson, Velocity dispersions and mass to light ratios for elliptical galaxies, Astrophys. J. 204 (1976) 668.
4. N. Kaiser and G. Squires, Mapping the dark matter with weak gravitational lensing, Astrophys. J. 404 (1993) 441–450.
5. D. Clowe, A. Gonzalez and M. Markevitch, Weak lensing mass reconstruction of the interacting cluster 1E0657-558: Direct evidence for the existence of dark matter, Astrophys. J. 604 (2004) 596–603, [astro-ph/0312273].
6. W. J. Percival, S. Cole, D. J. Eisenstein, R. C. Nichol, J. A. Peacock, A. C. Pope et al., Measuring the Baryon Acoustic Oscillation scale using the SDSS and 2dFGRS, Mon. Not. Roy. Astron. Soc. 381 (2007) 1053–1066, [0705.3323].
7. R. Dave, L. Hernquist, N. Katz and D. H. Weinberg, The Low redshift Lyman alpha forest in cold dark matter cosmologies, Astrophys. J. 511 (1999) 521–545, [astro-ph/9807177].
8. B. Paczynski, Gravitational microlensing by the galactic halo, Astrophys. J. 304 (1986) 1–5.
9. K. Griest, Galactic Microlensing as a Method of Detecting Massive Compact Halo Objects, Astrophys. J. 366 (1991) 412–421.
10. EROS collaboration, T. Lasserre, Not enough stellar mass machos in the galactic halo, Astron. Astrophys. 355 (2000) L39–L42, [astro-ph/0002253].
11. D. P. Bennett, Large Magellanic Cloud microlensing optical depth with imperfect event selection, Astrophys. J. 633 (2005) 906–913, [astro-ph/0502354].

...

14. K. Griest, A. M. Cieplak and M. J. Lehner, New Limits on Primordial Black Hole Dark Matter from an Analysis of Kepler Source Microlensing Data, Phys. Rev. Lett. 111 (2013) 181302.
15. P. Pani and A. Loeb, Tidal capture of a primordial black hole by a neutron star: implications for constraints on dark matter, JCAP 1406 (2014) 026, [1401.3025].
16. M. Milgrom, A Modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis, Astrophys. J. 270 (1983) 365–370.
17. C. Kraus et al., Final results from phase II of the Mainz neutrino mass search in tritium beta decay, Eur. Phys. J. C40 (2005) 447–468, [hep-ex/0412056].
18. V. M. Lobashev et al., Direct search for mass of neutrino and anomaly in the tritium beta spectrum, Phys. Lett. B460 (1999) 227–235.
19. E. W. Kolb and M. S. Turner, The Early Universe, Front. Phys. 69 (1990) 1–547.
20. S. D. M. White, C. S. Frenk and M. Davis, Clustering in a Neutrino Dominated Universe, Astrophys. J. 274 (1983) L1–L5.

Also see: "The mass of the dark matter particle from theory and observations" (10 Apr 2012), by de Vega, Salucci, and Sanchez, on page 12:

"9. Conclusions

Dark matter is characterized by two basic quantities: the DM particle mass $m$ and the number of ultrarelativistic degrees of freedom at decoupling $g_d$ (or, alternatively the decoupling temperature $T_d$). We obtain the density profiles and theoretical relations between $m$ and $g_d$ involving the observable densities ...

From the observed values of the surface density we present here clear evidence that the mass of the DM particle is about one or two keV.".