Note: Although I have heard the term "stand up philosophy" and "stand up philosopher" in other places, I haven't seen it in quite these terms.
Stand up philosophy, to me, is sort of street philosophy, where someone will stand up (or raise their hand), and expound a very deep, meaningful, and quite interesting chunk of thought. Often it is old (grandparent style) people who don't realize they are blabbing something useful among a wave of stories and anecdotes.
I think that conversation can be grouped into 4 categories: mechanics, abstract mechanics, bullshit, and philosophy, that is, everything else. Mechanics is the basic "pass the salt", day-to-day, completely necessary speech, with no hope of ever having deeper meaning. Abstract mechanics is the instruction/conversation about more abstract topics, say, discussing how to write an essay, lecturing about a topic, etc. It is often through these conversations that we stumble upon stand up philosophy. Bullshit is the utterly useless filler conversation, with no mechanical necessity, no transferral of ideas: basically wasted air. An example of this is a discussion of the weather. It is often difficult to distinguish between bullshit and philosophy (is that a stand up philosophy right there?), but philosophy is where something meaningful is discussed, but still has little/no mechanical relevance. Although the edges are blurry, bullshit and philosophy are separate beasts.
"Isn't it interesting that the whole world we live in, is actual cooled-down hellfire? Isn't that weird to think about? It says a lot about sin, and who we are as members of that world."
"Technology really isn't and end unto itself (technology for the sake of technology), the internet bubble burst years ago, that was about technology for itself. You'll see, in a hundred years nobody will even think of technology as anything on it's own, just like nobody cares about pens and brushes, but people care a great deal about writing and art." -- Jeff Robin (My art teacher)
It is important to think about something else, that someone's random ramblings may not have great meaning, and only by accident to they actually come across as being really cool. For example, the Greek prophets (is that the word? I mean the women who hung out in sulphur caves) never said a single coherent sentence, and yet the random babblings were interpreted any number of ways. Alternatively, it may not even strike someone that something is really deep, but others will.
It's also possible that the person philosophizing may not really have thought it over, and is just musing out loud (like me), and it comes over as random anecdotes and life stories.
Here are some of my Stand Up Philosophizing:
"What do the "fundamental questions" matter? The afterlife... when it actually starts mattering (when we die), we'll find out whether or not it exists!"
"Who cares?? We're all gonna die anyways, why not enjoy yourself?" Note: this one is a time-honored stand-up and ordinary philosophy principle, often known as existentialism
A little less stand-up, more thought through, although still thought of during a conversation:
"Existentialists traditionally believe that anything that's fun goes, from murdering someone to taking drugs. I don't think this works quite perfectly into my version of 'existentialism.' For me, happiness is not only in-the-moment, but also long term. Taking drugs and killing people, although it might/might not increase momentary happiness, will shoot long-term happiness in the foot, so I don't do them. I also believe that everybody else has their rights to be happy, so I don't kill them. Guaranteeing my long-term happiness is why I put up with bullshit like grades, homework, money, and any other idiotic bureaucracy that I wouldn't put up with, if I were going for momentary happiness."