Kurt Vonnegut loved his randomness, I see Diceman had been mentioned, also American Psycho and a lot of Ellison's earlier stuff when characters act on a whim - whether that's randomness or not that's up to you to decide.
RIBBBbert Anton WiILSOSNNNN is alsadadd chjekdkdf out too. Sorry I had a chaos storm just hit me.
William Holdstock Where the Time winds blow is good too.
It would be a mistake to be drawn too deeply into the semantics of "random" as it is one of those words that has meanings varying from how it is used in a technical mathematical context, right through to be a vernacular term for "unexpected and a bit odd."
Rather than defining what the word means, perhaps the more important question is what the OP meant when they used it.
I suspect they intended to mean something along the lines of a "chance event or encounter, outside the usual run of events, and of a kind that could not have been predicted given what was known before."
The problem then is that such random events are very common, but are seldom noticed because they have no obvious consequences for us. We like to construct narratives about our lives. We look for explanations, and for remarkable coincidences — but those coincidences are only defined retrospectively.
If I nod and say "hi" to a stranger at the bus stop, and he says "Hi" back, this is a random encounter that I will quickly forget. If he happens to ask me which stop he needs for a particular destination, I will answer his question, but soon forget the conversation — unless perhaps his destination is near to where I used to live, or is the same place that I am going to. We may then fall into casual conversation which might be enjoyable, but will be forgotten soon after we part company.
Now imagine that the stranger mentions the name of the person he is going to visit, and I happen to know that person, I may find that I have something in common with the stranger beyond the normal pleasantries. With a shared acquaintance, it is not unlikely that we may meet later at some social event, and this may lead to a lifelong friendship, all because of a random encounter.
Now imagine the person who takes the same bus every day, and gets off at the same stop at the same time every day. One day, a car goes out of control and hits the bus stop just as they step off the bus, injuring them or killing them. This is a sad coincidence, but in many ways unremarkable.
But take either of these two examples, and suddenly it becomes a remarkable random event that changed the course of their life:
They were late leaving work and for the first time ever they took the later bus, reached their stop 20 minutes later than usual, and were injured when the crash happened.
They were late leaving work and for the first time ever they took the later bus, and as a result they were not there when the crash happened, although several people they knew who had caught the usual bus were killed.
In either of these cases, it is natural for us to create a narrative:
If the printer hadn't jammed, and I'd been on my usual bus, I'd never have been in that accident.
The printer jamming saved my life — because if I'd left at the usual time, I'd have been involved in the accident.
Most of life is a series of coincidences and circumstances, with only a small part of what happens to us being caused by decisions that we make. It may not feel like that, because we like to feel in control of our actions, but in reality, most of us spend most of our time acting out of habit, instinct and reacting to our surroundings.
When I went on a dating website, I was presented with something like 20,000 possible matches. Within a week or so, I had met the girl who became my wife. Coincidentally, we were each about 1 year out of bad relationships and had gone onto the dating site only days before we saw each other's profiles. I had chosen that site out of all the dozens available, only because a friend had used it with some success.
If my wife or I had gone on that site a few days earlier or later, or I had randomly chosen a different profile to read out of the 20,000 available, we may never have met.
However, my rational mind tells me we would probably have each met someone else and been approximately as happy. Yes, a series of coincidences led to the life-changing fact that we married, but a different set of coincidences would have led the the life-changing fact that I would have met someone else: perhaps an equally ideal spouse, or perhaps a psycho who stole all my money and stabbed me in the eye with a pencil. Who knows?