I could've sworn this (or some variant of it) started as an expression in hang out culture sometime around the late 90s or early 2Ks. I remember a friend of mine using some variation of "so random" decades ago, usually whenever recounting some over the top, crazy humorous event that happened, usually involving something someone did while drunk, or some random "character" who showed up unexpectedly while they were hanging out and it was memorable to them somehow—usually, more memorable to the people who experienced it (often while they were drunk or high or both) than to the people hearing about it later on. You had to have been there to really get how "oh, so random!" it was.
I often hear women use that expression as well when remarking about some crazy thing "que random!" ('How random" in Spanglish).
I could see certain occurrences that happen in play as being "Oh, so random!", like when someone does something unexpected that sets the rest of the group laughing with incredulity. But I'd rather that sort of spontaneous occurrence be the only time that "Oh, so random!" things happen, rather than it becoming a standard part of play or something that people purposefully incorporate into their characters, like they're trying (too hard) to be "wild" or funny.