In the standard model the Higgs boson gives the mass to other particles, but in the Universe we know that the 80% of mass is in form of dark matter, that is not constituted by known particles. The Higgs boson gives the mass also to the dark matter with the same mechanism?
The question is whether dark matter gets its mass from the Higgs field. The answer depends on the composition of dark matter, so let's discuss the mass explanations for several common composition hypotheses in turn. (We needn't discuss alternatives to dark matter, such as MOND or gravity no longer obeying an inverse square law over kiloparsecs.)
If dark matter is neutrinos, blame the seesaw mechanism. (However, as a comment below notes, this hypothesis hasn't aged well.)
If dark matter is composed of MACHOs - in short, a particular kind of star or former star - almost all the mass comes from baryons (particles such as protons and neutrons), and almost all of their mass comes from the potential energy of the strong force holding the quarks together in baryons. This is why, although the Higgs field gives the quarks in protons some mass, the proton mass is dozens of times what would be expected from that alone.
Finally, if dark matter is composed of the lightest "supersymmetric partner", the Higgs field is responsible. One motivation for postulating supersymmetric partners much more massive than familiar particles is that it allows us to reduce quadratic divergences in the Higgs field to logarithmic divergences. One version of this theory conserves a multiplicative charge called R-parity, which is $1$ for known particles but $-1$ for their supersymmetry partners. The lightest such partner therefore cannot decay, even if its Higgs-induced mass is very large.
Dark matter is a necessary hypothesis within the general relativity model of the universe in order to fit the observational data of rotational curves. In simple words, the trajectories can only be explained if there exists a lot more matter in the galaxies than luminous matter. Luminous matter means that electromagnetic interactions generate light which is measured and used to calculate the amount of matter giving that luminosity. Dark matter must be composed out of masses that do not interact electromagnetically to first order.
There are various models that propose various ways massive particles could exist , for example MACHOs.
Massive astrophysical compact halo object (MACHO) is any kind of astronomical body that might explain the apparent presence of dark matter in galaxy halos. A MACHO is a body composed of normal baryonic matter that emits little or no radiation and drifts through interstellar space unassociated with any planetary system. Since MACHOs are not luminous, they are hard to detect. MACHOs include black holes or neutron stars as well as brown dwarfs and unassociated planets. White dwarfs and very faint red dwarfs have also been proposed as candidate MACHOs.
In this model the Higgs mechanism does not play a special role other than the usual standard model role in giving masses to its elementary particles.
There are elementary particle models, beyond the standard model, where stable weakly interacting massive particles exist in the spectrum, called WIMPs:
In particle physics and astrophysics, weakly interacting massive particles, or WIMPs, are among the last hypothetical particle physics candidates for dark matter. The term “WIMP” is given to a dark matter particle that was produced by falling out of thermal equilibrium with the hot dense plasma of the early universe, although it is often used to refer to any dark matter candidate that interacts with standard particles via a force similar in strength to the weak nuclear force.
For these particles there exists a Higgs mechanism which will give them mass as elementary particles of these new theories.
Please note that it is the Higgs mechanism, the Higgs field that gives mass to the elementary particles in the standard model, not the Higgs boson. The Higgs boson is a predicted manifestation of the existence of this field, and acquires its mass from this field too.