My opinion: Words such as electrons and neutron are just shorthand to sum up a collection of physical properties (including mass) that each different particle in the standard model possesses. What an electron really is, obviously I don't know.
In time I presume the SM itself will be modified, but the SM is not really a collection of the actual 100 percent certain, building blocks of the universe. It is a list of physical properties, such as mass and charge.
Physics is not about what things "really" are. It's about learning as much as possible about the properties that these (currently considered) elementary particles possess or exhibit, both by theory and experiment and then making predictions that help refine our understanding of these properties.
For example, if we discover the particles that possibly make up dark matter, or the forces that cause dark energy, physics will still be in the dark as to what reality and matter actually "are" but that's not what physics is designed to find out.
It is a science based on the properties of things, not the things themselves. We may never know if an electron is an entirely mathematical concept, or is there a tiny nugget of something "real", that we can relate to the classical world. We can only measure a property or effect of an elementary particle, that is as much as we can say about it.
To me you are asking a philosophical or ontological question, but not a physics question. I say this because, by "matter" you mean something that we can deal with in classical terms, but as long as physics sticks to the properties we can measure, we don't really have to worry about what actually produces the properties.
I am sure you know this bit, but please bear with me. Take a piece of matter, say a lollipop stick, and start cutting it up. Eventually you are left with atoms that you can pull the electrons off of, by ionisation. So the atom is the last /smallest bit of matter.
This is where it gets odd, because you can measure all the aspects of that electron, but you can't tell what it really is, that is a "thing" with these properties:
Able to have a basic electric charge that can't be added to or taken away from,
You need to spin it 720 (not 360) degrees to get it back to where it started (so it's not a tiny, tiny soccer ball, and it can be in many places at once, when you are not looking at it). Spin does not literally mean revolving, it is a holdover from 100 years ago when they imagined an electron as an tiny ball.
It's "spin" is fixed, you cannot spin it up or down, faster or slower.
You cannot tell in any way or by any test, one electron from another. The list of odd things goes on, and the further we get from the definition of matter we were both told about in school.
The main point is that we think, because we haven't got the technology, so we can't acually measure it's "size", that it is a dimensionless point, whatever that is :).
So this is where our high school definition of matter might as well be dumped, because whatever an electron is, it's not everyday matter anymore. It's the same with the other bits of the atom.
So, as far as I know, (because there are people here much more knowledgeable and experienced in this than I am), you have three choices.
$E=mc^2$, so you can call it energy, because it looks like energy at small scales and looks like mass/matter when you get enough of it together to make a loppipop stick. We don't have a good definition of energy, which does not help.
You can say, well ordinary rules and ordinary words can't describe it, but math (as a seperate language) definitely can and does describe it very well. So because of that you might argue, that the universe (and us) are actually mathematical in origin, since it follows math rules and can only be described by math.
You can turn to a "prime mover", (this is meant to include all sorts of intelligences beyond ours, if they exist :) and treat the idea that we don't know what exactly an elementary actually is, as being in the same league as trying to figure out other hard problems, such as how/why can we worry about this in the first place, (and build cars, houses and all the other things humans can do but other animals can't).
It other words, I suppose, we would have to accept the fact that we are just not smart enough to figure it out any further.
My personal preference is for number 2, but it sure is a long, long way from the "common sense" view of matter we learnt in school.
You asked about GR, and this is another mystery that we would like to get sorted out. The equation that defines mass in GR gives the same value as the equation that defines mass in equations describing inertia. There is no obvious reason why these two masses shold have the same value, but they do. I would refer you to Principle of Equivalence
EDIT My sincere apologies, I now realise the above wandering and repetitive piece was written for myself, to clarify basic aspects of physics that are blindingly obvious to most other users. Thank you very much for asking the question, it helped me understand the role of physics. END EDIT.