I can’t remember where I read this, but I’ll post it anyway (not verbatim).
The reason why we say kaddish (or daven for the omud or learn) for a niftar is to replace the kiddush hashem that has been lost since that persons passing. The best way to replace that kiddush hashem is to learn torah. As not every one was capable of learning (no artscroll back then) they would daven for the omud. The problem arose as to what to do for a koton who could not learn and was not able to daven for the omud until his bar mitzvah. It was for these children that the saying of kaddish was instituted.
Kaddish being said by a mourner etc.. is first mentioned by the Ohr Zarua in the 13th century.