This post mostly consists of my notes and beginning research on the subject.
Here are my thoughts:
Rock-Paper-Scissors and building on that. Image from here.
Sploder allows you to make your free games online. Don't know much about it.
Wiki-how suggests ways to make your own board game and trading card game. MIT Game Lab offers a 50 minute video on how to make a card game.
I offered a quick-and-dirty version of a Role-playing game:
The Cat Kingdom of Sam-hwan Apartment Complex
I lived in Sam-hwan Apartment Complex for a few years when my family moved to Busan and the map is from Daum - Korea's main Internet Portal - with some green and brown bushes added by me. The half-formed idea is that cats must travel the complex looking for wild prey and garbage. Cuter cats get fewer stones thrown at them by humans while smelly cats have a different defense... How to win or gain experience points wasn't decided. Well, a lot wasn't decided but this was enough to wing a few rounds exclusively using a six-sided die.
A colleague has ideas for making board games as well.