Out of curiosity I looked up その in a monolingual dictionary, the first thing I see Is ［連体］《代名詞「そ」＋格助詞「の」から》What on earth is a "case marking particle"? Is this the の particle that simply means "of"?
So, if I were to translate it literally, a phrase like その上 would mean "above of that" "on top of that" while "その人" "person of that" Sounds like complete gibberish to me when I try to think of it that way though.
A 格助詞【かくじょし】 or "case-marking particle" is a particle that attaches to a noun form to mark how the word relates to the rest of the sentence or clause. There are several, including ～は, ～が, ～を, etc.
～の is a confusing one because there are several different particles all pronounced 「の」, but in this case, I believe it marks the genitive case, generally used for possession.
In this particular case, the given analysis of 《代名詞「こ」＋格助詞「の」から》 is only applicable to Old Japanese, as 此【こ】 -- originally a pronoun (代名詞【だいめいし】) -- is no longer used by itself.
Now, この, その, あの, and どの are simply fully-fledged pronouns. (Incidentally, this is all basically true of the other suffixes originally attached to these pronoun bases, including これ, ここ, どっち, etc.)
Here's the skinny. あ, こ, and そ are each technically pronouns on their own. Usually we see them in conjunction with a particle. In this case, it's connected to the case marking particle (格助詞) "の". A case-marking particle is a particle that indicates the grammatical case, or basically the function of the thing it's modifying. So for example の can mark the genitive case (私の本 = my book). に can mark the locative case (日本に住んでいる = I live in Japan). を can mark the accusative case (ボールを投げた, where ball is the direct object). The wikipedia page talks about all the cases pretty clearly, and if you want to venture to the Japanese page on 格助詞 you can. At the very least you can see a list of the case marking particles.
So in あの・この・その we have our pronouns plus our case marker の. This forms what is called a 連体詞{れんたいし}, or a prenominal adjective. As you may be able to guess, it's an adjective that comes before a noun. This ~の form of 連体詞 is basically its own class that is the form of noun + 格助詞「の」. As per wikipedia: "本来は「名詞」+格助詞「の」だったものが多い。"
That might be a little more technical than what you want to hear. A shorter version is that yes, it is the "の that means 'of'," but you may want to stop trying to think of it in terms of its rough English translation, because it's exactly that: rough. It does not have the exact meaning of "of." You can think of it in terms of having that genitive kind of possessive-ish meaning, but try to separate it from English if you can. その人 isn't "person of that," but it's the pronoun そ which is modifying 人 in the genitive case. So which person? That person, the one I'm connecting to そ. In English we usually refer to の as a 'noun modification' particle, at least in educational materials I've seen, so try thinking of it like that. Sorry if it's hard to make the connection, but I hope you get it!
Ultimately, this grammatical distinction isn't that important. You won't see そ or あ or こ on its own as a pronoun anywhere, and I would assume nobody aside from linguists would actually think of them in terms of their constituent parts. For all intents and purposes, they are their own words that just happen to have a more grammatical history.
