If I understand correctly, we’re not yet able to explain exactly why massive objects attract each other. Some theories posit the existence of gravitons, but those are still only hypothetical (and not all the current theories require them anyway).
A different area of research is giving us strong evidence for the Higgs boson and how it imparts mass to objects.
Since gravity arises due to an object’s mass, does the Higgs have any implications for our understanding of the way gravity works? Does the Higgs make gravity easier to understand, or does it raise new puzzles?
Or do they have nothing whatever to do with each other?
We would normally say that we do understand why massive objects attract each other: General Relativity is a mature and successful theory. Gravitons are expected from trying to combine the lessons of Quantum Mechanics to General Relativity.
Your question about the Higgs and gravity is a good one. But the answer is that that they are not related. Gravity results from energy (technically the stress-energy tensor, but I’m trying to stay non-technical), not mass. Mass happens to have a lot of energy (E=mc^2), which is why it is a source for gravity. The Higgs boson does not give particles their energy. It does not create energy. All it does is bind energy up in the form of mass. For this reason it has nothing to do with Gravity.
Just to clear up several misconceptions at once:
1: The Higgs boson itself isn’t responsible for giving anything mass. That’s the Higgs field. The Higgs boson is just an excitation of the field, and detecting those excitations is evidence in favor of the Higgs mechanism, but it’s not directly involved in it. By way of analogy, we can think of the Higgs field as a trampoline, and assume that the trampoline is responsible for something important. If you thwack the trampoline in the right way, then you can make ripples in it (the ripples being analogous to the particle). If you see those ripples, then you know that the trampoline exists, but the ripples aren’t necessary for the trampoline’s existence.
2: The Higgs boson isn’t responsible for all mass. The Higgs mechanism applies only to fundamental particles, not composite ones, and probably not even to all fundamental particles (the neutrinos may well instead get their mass through a completely unrelated mechanism). So the Higgs isn’t responsible for mass as a whole, just some specific examples of mass.
3: The Higgs boson isn’t even responsible for most of the mass we’re familiar with. Most of our mass comes from protons and neutrons, and protons and neutrons are made up of quarks, and the masses of the quarks come from the Higgs process. But the mass of a proton or neutron is much, much higher than the sum of the masses of the three quarks that make it up: Most of the mass of a proton or neutron comes from the strong-force binding energy that’s holding those quarks together. And thus, most of our mass likewise comes from that strong-force binding energy.