How literally should you take "The Higgs boson gives other particles mass"? A standard phrase in popular discussions of the Higgs boson is that "it gives particles mass". To what extent is this a reasonable, pop-science, level of description of the Higgs boson and it's relationship to particles' masses?
Is this phrasing completely misleading?
If not, what would be the next level down in detail to try to explain to someone?
 A: A standard simple answer (for the standard Higgs boson field) is that a particle acquires mass by passing through this field, which changes the particle's inertia (thus appearing as acquiring mass which is a measure of inertia among others)
Of course the standard Higgs boson is still investigated (if it is the standard one and not some variation of other proposals) and this is not the only way one can reason or explain how it gives mass to other particles.
A: You should take it completely literally. (Quibbles about the Higgs field vs the Higgs boson are misguided. Particles don't acquire masses until the point at which the Higgs boson appears, so attributing the particle masses to the Higgs boson is just as correct.)
However, there is a simple way to picture this. The concept of a Higgs boson is completely generic, although in particle physics, one is usually referring to the standard model Higgs. A "Higgs boson" appears when a system undergoes a phase transition that breaks some symmetry.  
There are many examples from solid state physics where the same thing happens. In a superconductor, for instance, at the critical temperature, the cooper pairs become the "Higgs" bosons and the particle which acquires a mass is the photon. This is the famous BCS theory of superconductors. Compare with Ginzburg-Landau theory of phase transitions, in which one expands the potential in even powers of the field).
A: Short answer: do not take it literally, without further context.
In order to understand the Higgs boson's role in the Standard model, it is necessary to take a closer look at the framework in which we describe elementary particles: quantum field theory. 
In this approach, particles are described as excitations of fields that spans all spacetime. The ground state of the field corresponds to vacuum, what we call particles corresponds to excitations of the latter. If you are familiar with quantum mechanics, think of a harmonic oscillator to understand the concept. 
The point is now that the mass creation effect is due to the presence of the field, not the associated particle. The existence of the particle arises as a kind of consistency requirement, which was confirmed recently at the LHC. 
In this sense, the answer to your question depends entirely on what you mean by "Higgs boson": having massive particles does not mean that there are bosons in the sense of particles constantly surrounding them. They are far too heavy for this to be a viable option, as 125 GeV is far beyond what one experiences in daily life (a proton comes at a rest mass of almost 1 GeV). 
For a particle physicist, it is obvious that "boson" refers to a property of the field as a whole, and not the excitation of the latter. A layman, however, will associate it with a particle moving through space time. Hence, you should not take this phrase literally. I would refrain from using it freely without providing any additional explanation.    
A: "Binding a massless particle into a small space" is a good phrase for a popular discussion, but it is not the only way to picture the Higgs mechanism.
Another perspective comes from the fact that every particle inside some interaction field behaves exactly like its energy or momentum has changed. This concept is called canonical momentum, in contrast to the usual (kinetic) momentum $m\vec{v}$. For example, in the magnetic field the canonical momentum is $\vec{P}=m\vec{v}+e\vec{A}$, and in the static electric field the canonical energy is $E=\tfrac{1}{2}mv^2+e\varphi$ (these formulas are non-relativistic). The latter is most simply understood, because we are used to call $e\varphi$ potential energy. A force acts on the particle when such additional term changes with spatial position.
Variations of this idea depend on the tensor type of the interaction field. The electromagnetic field is a vector field, and the Higgs field is a scalar field. (The other types of fields are also possible, for example, the gravitational field in GR is a tensor field of order 2.) That leads to an important fact: the energy and the momentum change by the same factor (in the relativistic sense), which is the same as the mass would change. $\vec{P}=m\gamma\vec{v}+\Delta m\,\gamma\vec{v}$ and $E=m\gamma c^2+\Delta m\,\gamma c^2$, $\Delta m=gh$ where $g$ is the coupling constant.
Thus, wherever some scalar interaction field gets a non-zero value, particles move like they have gained some mass. And the Higgs field does have a constant non-zero value all over the Universe, $h=h_0$. Thus, all particles have their Higgs masses, and they can have no explicit mass besides it, and the theory supposes so.
A: The Higgs field (note it is the field that is important here, not the Higgs boson itself, which is just a ripple in the Higgs field) gives particles mass in the same sense that the strong force gives the proton mass (context: $99\%$ of the mass of the proton comes not from the mass of its constituent quarks, but from the fact that roughly speaking the quarks have a large amount of kinetic energy but are bound by the strong force). If any force confines energy into a small amount of space, then that bound energy has a mass given by $E=mc^2$. This is what the Higgs field does: it binds a massless particle into a small space, and therefore by $E=mc^2$ (and the fact that the particle now has a frame of reference in which it is stationary) that particle has an effective rest mass.  
To get an intuitive feeling for what's going on, as an exercise you can derive $E=mc^2$ by considering a photon confined by a mirror box. The photon is bouncing back and forth exerting pressure on the mirror, and if you try to push the box it will have inertia due to the photon exerting more pressure on the front of the mirror than the back. If you work it out you will find that the mirror box has an effective inertial mass of $m=E/c^2$. The Higgs field provides a force that acts like this mirror box, thereby "giving" mass to the particle inside it.
