If there was but one object in the universe, can it be accelerating or not accelerating? If there is only a single (material) object in the universe does it still make sense to speak of it as accelerating or not accelerating?
I believe it might be an equivalent question to ask whether it makes sense to speak of two objects in the universe and no more, which are accelerating or not accelerating in exactly the same way.
(I am not worried about issues such as: what would exert a force on it? I am interested in, as it were, the concept that underlies that of acceleration.)
BACKGROUND
I am sorry to be asking such a basic question.   From this answer to another post I gather that even if two things exist in the universe talk of acceleration makes sense.  (In that answer, the two objects are one person and the rest of the universe.)  But what if only a single thing existed?
Notice in that answer why the idea of "the rest of universe's accelerating" is being invoked.  It is to break the symmetry in the Twin Paradox. The impression one gets there is that acceleration (accelerating-ness) is an attribute of an object that it may have without reference to any other object. I am trying to get a better hold of the status that acceleration has in modern physics.
Or see the first (and so far only) comment to this post, which suggests that it makes sense to speak of one of the only two things in the universe as accelerating but not the other.
Even an answer or comment that points me to appropriate search terms would be highly appreciated.
 A: 
If there was but one object in the universe, can it be accelerating or not accelerating?

The answer depends on the meaning of "object." In general relativity, one of the dynamic entities is the metric field, which 


*

*mediates gravity, 

*defines the distinction between timelike and spacelike directions, 

*determines which motions qualify as inertial ( = free-fall = weightless) and which ones don't. 
So if the universe had only one object together with the metric field, then the distinction between accelerating (non-weightless) and inertial ( = free-fall = weightless) would still be meaningful, at least if classical general relativity is used as the basis for answering this hypothetical question. 
A similar comment applies in special relativity, the only difference being that in special relativity the metric field is fixed rather than dynamic. When we write an equation like
$$
 d\tau^2=dt^2-\frac{dx^2+dy^2+dz^2}{c^2}
\tag{1}
$$
for the proper time $\tau$, or 
$$
 ds^2=-dt^2+\frac{dx^2+dy^2+dz^2}{c^2}
\tag{2}
$$
for the proper distance $s$, we are implicitly specifying the metric field. It doesn't mediate gravity in this case (because it's fixed), but it still defines the distinction between timelike and spacelike directions and still determines which motions qualify as inertial. 
Equation (1) or (2), which specifies the Minkowski metric, is part of the foundation for special relativity. With this particular metric in this particular coordinate system, a timelike world-line represents inertial motion of a pointlike object if and only if it is "straight", meaning that $x,y,z$ can all be expressed as linear functions of $t$ (constant velocity). If "accelerating" is used in the OP to mean non-inertial ( = not weightless = not in free-fall), then we don't need more than one object, unless we count the metric field itself as an object — which could be a legitimate generalization of the word, because it is a dynamic entity in general relativity. 
In a "universe" without a metric field (or anything to take its place), there would be no distinction between timelike and spacelike directions and therefore "acceleration" would be undefined no matter how many other objects were present. So, in the context of relativity, the question itself assumes the presence of a metric field, and then the answer boils down to a language issue: whether or not we count the metric field as an "object." 
The OP specifies a material object, and since "matter" typically implies protons and electrons, the metric field does not qualify as a material object the way we typically use the words. Then the answer to the question is yes: the distinction between accelerating (non-inertial) and non-accelerating (inertial) is still meaningful even with only one material object in the universe, as long as a metric field is also present.
A: I think, general answer is that no, it wouldn't make sense. We usually have at least two "things" when discussing motion: reference frame and the body we are studying.
We need to know in which direction the body is accelerating.
Real-life example: right now you are moving with pretty high velocity through the space (relative to some star), as well as being accelerated around the sun (albeit slightly). You don't know it, or notice it, unless you actually look at the sun and stars.
A: Accelerating and not accelerating are concepts tied to your reference frame.  This is a mathematical construction which a physicist constructs with which to solve problems.  Your acceleration can change with respect to these systems.  To explore this, we should start with a simple case and add complexity until we come full circle to your single object universe.
In a world with multiple objects, it is easy to see how this works.  If I have a bowling ball falling to the ground, I can make a reference frame such that the ground is holding still, and see that the ball is accelerating in that frame, and the ground is not.  I can also make a reference frame from the point of view of the ball.  In this frame, it is the ground that is accelerating towards the ball, and the ball is motionless.
In general, we find these to be the most convenient reference frames.  However, sometimes more exotic frames are useful.  For example, in the above example with the ground reference frame, we ignore the effect of the ball pulling on the Earth.  A bowling ball doesn't have much pull.  It will pull the earth nanometers at most as they come together.  But the earth does indeed move.  When studying astonomical objets, this matters.   Consider our solar system.  We like to say that Jupiter revolves around the sun, so it would make sense to create a coordinate system where the sun is fixed and Jupiter is in motion, right?  Well, almost.  If we did this, we'd see Jupiter "wobbling" for some not-fully explained reason.
The reason for this wobble is that Jupiter does not precisely orbit the sun.  Objects actually revolve around the center of mass of the whole system, known as the "barycenter."  In our case, Jupiter is actually massive enough that the barycenter isn't at the center of the sun.  Sometimes it's actually outside of the photosphere of the sun (though it never actually gets far enough to leave the corona).  If we create a new reference frame centered on this barycenter, we find the orbits are much simpler.  Both Jupiter and the sun orbit around the barycenter of the solar system in nice clean orbits (ignoring the very minor effect of all non-Jupiter planets).
But note that I just built a reference frame around the barycenter of the solar system.  The barycenter is not an object.  There's no diamond crystal at that point marking the barycenter.  It's just a mathematical concept.  I built a reference frame around a mathematical construct, rather than a physical one.  You're allowed to do that.
So going back to your one object system, I can build any reference frame I please.  I can make a reference frame which would follow the trajectory of some hypothetical rocket as it streaks off into space, whether or not that rocket actually exists.  The math which we use to predict the motion of things in our universe will work no matter what reference frame you pick.  And in that reference frame, our one object in the universe will indeed be accelerating.
Now such exotic reference frames would typically not be considered practical.  I would expect any reasonable practical problem would end up constructing a reference frame centered on the one object in the universe, because that creates a symmetric situation and it's much easier to do math when there is nice clean symmetries.  Anyone using a difference reference frame will probably be asked to justify their choice, but the math will still work.
