How does Higgs Boson get the rest mass? Higgs Boson detected at LHC is massive. It has high relativistic mass means it has non-zero rest mass.
Higgs Boson gives other things rest mass. But, how does it get rest mass by itself?
 A: The mass of the Higgs boson is a free parameter of the standard model and not (only) due to the interactions with a non-zero Higgs field.
If the Higgs field were zero, the standard model predicts four massive Higgs bosons, which are the only massive particles. In case of a non-zero Higgs field, only one of them gains some extra mass via Higgs field interactions and becomes 'the' Higgs boson (assuming that there is only one), while the remaining Higgs bosons mix with Isospin and Hypercharge gauge bosons to form the electroweak gauge bosons.
For further information, see this article by Matt Strassler.
A: You have to read a bit about the Higgs mechanism. 

In particle physics, the Higgs mechanism (also called the Brout–Englert–Higgs mechanism, Englert–Brout–Higgs–Guralnik–Hagen–Kibble mechanism,1 and Anderson–Higgs mechanism) is the process that gives mass to elementary particles. The particles gain mass by interacting with the Higgs field that permeates all space.

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The simplest implementation of the mechanism adds an extra Higgs field to the gauge theory. The spontaneous symmetry breaking of the underlying local symmetry triggers conversion of components of this Higgs field to Goldstone bosons which interact with (at least some of) the other fields in the theory, so as to produce mass terms for (at least some of) the gauge bosons. This mechanism may also leave behind elementary scalar (spin-0) particles, known as Higgs bosons.

*This left over spin 0 is the Higgs * that we hope has been discovered, if nature follows the simplest  higgs mechanism implementation. It acquires its mass by the Higgs mechanism too.  That is why the scientists are careful to state that more work is needed to establish what type of Higgs particle  this is. 
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The Higgs mechanism was incorporated into moder particle physics by Steven Weinberg and Abdus Salam, and is an essential part of the standard model.
In the standard model, at temperatures high enough so that electroweak symmetry is unbroken, all elementary particles are massless. At a critical temperature the Higgs field becomes tachyonic, the symmetry is spontaneously broken by condensation, and the W and Z bosons acquire masses. (EWSB, ElectroWeak Symmetry Breaking, is an abbreviation used for this.)
Fermions, such as the leptons and quarks in the Standard Model, can also acquire mass as a result of their interaction with the Higgs field, but not in the same way as the gauge bosons.

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A: The Higgs gives the W's, Z's, quarks, leptons, and (probably) the neutrinos a mass, but it doesn't have hardly anything to do with the proton mass, it only contributes the tiny amount roughly equal to the proton neutron mass difference. Its own mass is fundamental parameter in the standard model, and the natural value is the Planck mass.
Physicists call the question of why the Higgs is so much lighter than the Planck mass "The Heirarchy problem".
A: Forget about relativistic mass; it's an outdated and, in this case, irrelevant concept. The Higgs boson has a rest mass of about $125\ \mathrm{GeV}/c^2$ assuming it is in fact what the LHC has found.
Anyway, I would say that the Higgs boson does not actually give other particles mass directly; instead, it's a side effect of the mechanism by which those other particles become massive. It just naturally turns out that the particle produced by this mechanism has to be a massive particle itself.
Or to put it another way, the Higgs field would not be able to give other particles mass if it were not itself massive. Take a look at the "Mexican hat" potential shown in this site's logo. The bump in the middle arises because the Higgs field has an associated mass, the mass of the Higgs boson. That bump pushes the "natural" state of the Higgs field off center, which means the field has a nonzero "default" value, called the vacuum expectation value. It's that vacuum expectation value that gives other particles mass. Without the bump, the minimum of the potential would be in the center, which means the vacuum expectation value of the Higgs field would be zero, which in turn would render it incapable of giving other particles mass.
I'll refer you to another answer of mine for some of the mathematical detail.
