I feel like I should take a swing at attempting to answer this question, but I have to tell you, it is a tough one! People who do not like or do not "get" abstract art, frequently say things like, "My five year old could have done that." Well.... yes and no. The act of putting paint on a canvas or pressing the shutter on a camera is easy; anyone can do it. But what is the product?
As I've mentioned previously, I tend toward the engineering, objectivist world-view. That can be very binary. Something works, or it doesn't. Something is good, or it is bad. That kind of thinking does not apply in general to art and, in particular, to abstract art.
So, what makes good abstract art? I believe there are several factors to be considered. (This, by the way, is my personal view. Not all artists share it.) First, let's talk about impact. There are pieces of art that, when you walk by, you stop and say, "Wow!" I would suggest "Guernica," by Picasso, is a good example. Have you ever seen it? I can pretty much guarantee it will stop you in your tracks. You may or may not understand what he is getting at, and it may or may not be to your taste, but it definitely has impact. How about "Water Lilies" by Monet? Definite impact. The Impressionists, like Monet, were trying to present the feeling of looking at a scene rather than a photorealistic depiction. Or take "The Starry Night" by Van Gogh. Is that an exact representation of what he saw? No, of course not. It is abstract art. It has impact.
Another way to consider a piece of art is what I call the "Capitalist" way. What are people willing to pay for the piece of art? If one piece sells for $50 and another for $50 Million, that gives you an indication that many people feel the second one is good. Does that actually make it "good"? Not necessarily, but it is something to take into consideration.
Then there is the purely technical view: Are the various elements of the work in balance? Are the colors harmonious or, conversely, are they deliberately jarring? Where does the viewer's eye go when looking at the work? In most good abstract artwork, your eye will follow a path through the piece. The human eye is attracted by light colors more than dark, so your eye will tend to move from light area to light area. Does your eye move in a smooth curve or line, or does it jump haphazardly all over the piece? If your eye moves smoothly, the artist has probably taken the viewer's perspective into account, rather than just slapping some stuff into the scene (as your 5 year old might have done!).
All these aspects, along with individual taste, contribute to whether a piece of art can be considered "good" or not.
There is another area of abstract are that probably mostly appeals only to people who are really into abstract art. Certain artists have played with visual perception. Some colors, when placed next to each other, cause a shimmering effect. Mark Rothko is known for this. Some artists have played with lines that form patterns that can make the viewer dizzy or can seem to shift from one shape to another. Then you have artists like Jackson Pollack who used dripped paint to create his works. I'll be honest. I don't get Pollock's work. But a lot of people do. If you are someone who gets it, you probably did not need my explanation in the first place!
These comments have been focused on abstract art. but what about abstract photography? I recently saw a quote from William Neill (Outdoor Photographer magazine, December, 2019) that I like:
"Merriam-Webster's Collegiate Dictionary defines the word abstract, related to art, as 'having only intrinsic form with little or no attempt at pictorial representation or narrative comment' Photographic abstractions of nature are based in reality but composed to give no clear reference to it. With an abstract photograph, we know the subject is real. The mind wants answers."
I would mildly disagree and say that it does not necessarily give NO clear reference; some abstract art contains recognizable although not realistic images. If a photo leads the viewer to wonder what it is a shot of or leads the viewer to want to know more, then I would consider that to be an aspect of "good" abstract photography.
If you are new to abstract art or if you are puzzled by it, I hope my thoughts are helpful. Abstract art can be fascinating. Really! Trust me!