I actually liked "designing" games on paper when I was a kid, but most of them were sequels of games I liked. Like a super mario world involving a bobomb boss, and a super duper king k. rool that turned into super muscular, rather than being a fatso.
I did make a lot of comics inspired about video games characters, mostly about Clay Fighters and Earthworm Jim, up to like 10th garde

By that age, most of the plots and artwork where heavily inspired by Dragon Ball and DBZ, including super clayjins! Bad Mr. Frosty and The Blob were like a gay dynamic duo, now that I remember. At least the Blob was such a crybaby (much unlike its official incarnation). Octohead was more often than not the Big Bad.
I remember when I was like 7 years old I did a mini comic about the Clay Fighter blob, being formerly a human who was consumed by a green substance until he became the aforementioned character, way before the spiderman cartoon showed up the origin of spidey's symbiote (remarkably similar to my comic!). Damn plagiarism!
My brother and I used to do draw games on paper all the time. We started whenever our compulsive liar friend told us that sega would make your game if you designed all the levels on paper and mailed it into them. He also told us that a friend of his created bomberman that way.
Did anyone not grow up knowing that one kid who was a compulsive liar about video games? It's like some kind of archetype that transcends geography and culture.
Hey I was one, except it was more about movies than games. When a kid, never having seen the alien movies, but being a big fan of the toy line (they were awesome, I think I still have a few of them stashed somewhere), I made up the whole goddamn plot of Aliens (the sequel, if you must know) and I involved almost every single species concoted in the toy line, including the super badass scorpion alien and the snake alien. Fortunately, it seems I was good at making stories (at least to my target audience), for some of them found them cooler than the actual movies were, once we all watched them for the first time.
Oh, and I remember when I said the Exorcist, not knowing what exorcism actually meant, was about a serial killer (the titular exorcist... hey, it sounded like a title you'd be given for doing horrible shit when you were a kid), and gave you nightmares for months via subliminal messages.
But I did know a group compulsive videogame liars, particularly claiming shit about the existance of many bootleg games, like Megaman Z, which they claimed was a pirated Megaman X, but it was super awesome and Megaman had a lot of new moves.
Interestingly enough, I actually had the chance to meet a compulsive video game liar right at the first year of college, in a computer science related career. The guy told us that he could cheat LoZ: OoT to get the triforce and turn the game into spanish, and that he was super awesome and beat the last unbeatable epona race. Most of us were already pseudo romhackers and game modder, so most of us knew the boundaries of many games, and this guy spouting us shit at the age of 20 years was downright laughable. He was actually very stupid overall.
Oh, and does anyone remember those myths about Killer Instinct's B. Orchid's alternate finishing move?