John's right -- generally when CJ is used in a fic, it is short for Clark Jerome Kent, Jr, or in the case of alternate universe stories, a nickname Clark uses for himself. For their kids, Jonathan is also pretty common for boys, and I seem to remember a stretch where Lanie was popular for girls, as was Emma/Emmy (maybe after Teri had her daughter?) And, of course, the S5 & S6 teams chose Laura.
As for Clarkie-the-Bear, I can't remember when that one came on the scene, other than it was very early! So early, in fact, that I wonder if it might have been mentioned in a script or by the actors, but it could have just been named in a very early fanfic and it was so logical that it stuck. In fact, the first mention of it in a fanfic that I can recall came in the majordomo days, so you know it's been around a long time.
ETA: I was curious, so I went through my old computer files to see if I could find the story I was thinking of, that mentioned Clarkie the bear early on, and I actually found it. It was from 1995! And yes, it was from the majordomo days, before we had a website aarchive. Alas, it's no longer on the archive so I can't link to it, but for those old timers like myself who remember the salad days of the fandom, it was called "Clarkie: A Thanksgiving Story". (Now how's that for a blast from the past?

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Kathy