My language has several cases. I have their affixes worked out for the nouns, as well as the proto-affixes, but what I want to know is, how can I get the pronouns to have different-looking cases? I know I can just simplify (e.g. the dative was *-lū, and 1sg *īs; deriving the 1sg dative from *īlu instead of the proper *īslū gives modern yel vs. 1sg ye [properly yes]), but I want something more like she vs. them, or I vs. me. How could I do that?
2 Answers
To get a disconnect that extreme between different cases of the same pronoun, there's really only one tool for the job: suppletion.
Suppletion is the process where Word A gets reanalyzed as an inflection of Word B. For example, in English, the adjective good has the comparative and superlative forms better and best, respectively - but we would regularly expect *gooder and *goodest if good worked like all other adjectives, so how did this disconnect happen? Was it some horrifically complicated sequence of sound changes? Nope! Better and best descend from a completely different Proto-Germanic root than good. They were originally the comparative and superlative forms of an entirely different word, *bataz, until - for some reason, lost to time - *bataz fell out of use and was regularly replaced by *gōdaz (from whence good), but the comparative/superlative forms of *bataz for some reason didn't get replaced by the comparative/superlative forms of *gōdaz. Thus, *gōdaz snuck into the co-opted the inflections of *bataz, making them now irregular inflections of *gōdaz instead. Suppletion!
Pronouns can be susceptible to this too. As you mention, "I" and "me" look very different, very possibly the end result of some suppletion, but that wasn't English's doing, or even Proto-Germanic's. This weird suppletion-looking "I/me" distinction can be reconstructed all the way back in Proto-Indo-European, and got inherited into all its daughter languages. Since the result of (I'm assuming) suppletion goes back as far as the comparative method is capable of taking us, it's hard to say which form was originally what other word - they've been linked together for as long as we have data. You could do the same, and make different cases of the same pronoun look totally different as far back as the proto-language, and not bother to explain it beyond "well who knows why the proto did that; if we could reconstruct further back than the proto, then that would be the new proto ¯\_(ツ)_/¯".
If you insist on evolving suppletive pronoun forms though - it's telling that IE languages all seem to derive their 3rd person pronouns from demonstratives - words for "this one" or "that one". If you already have a word for "he/she/it", and then derive another one from a demonstrative, you can smoosh them together to get case suppletion. We can see this in action in Georgian: the 3rd person singular pronoun, in 6/7 cases, is derived from m(a)- (ergative მან man, dative მას mas, instrumental მით mit, etc.) which looks suspiciously like the ergative case ending; one is maybe derived from the other. But the nominative case is ის is... which also happens to be the distal demonstrative pronoun ("that one").
Hungarian took the word mag "body" and derived a number of reflexive pronouns from it, e.g. magam "myself", magunk "ourselves", etc. But the most important one is maga "him/her/itself", which acquired a dual meaning as a formal, polite 2nd person, non-reflexive pronoun. Later, Hungarian derived another formal 2nd person pronoun, ön, from a reflexive affix. They're both just polite ways to say "you", but deapite both being derived from a reflexive form, they fundamentally come from completely different roots. This hasn't happened in Hungarian yet, but you could imagine a scenario where, if ön is preferentially used in certain cases but maga is preferentially used in other cases, then they could be smooshed together into a single, suppletive pronoun with e.g. nominative ön, but accusative magát.
Hungarian is an interesting example - the (personal) pronouns are not inflected by the (agglutinative) suffixes, but vice versa, the suffix is inflected by the pronoun.
E.g. the dative suffix is -nek (simplifying a bit):
ember -> ember-nek, "to the human"
but:
- én (1st person singular pronoun, "I") -> nekem
- te (2nd person singular pronoun, "you") -> neked
- ő (3nd person singular pronoun, "he/she/it") -> neki
- mi (1nd person plural pronoun, "we") -> nekünk
- ...
you get the picture (this is different from Finnish).
Then apply repeatedly regular phonetic changes and case merging and you end up with a completely different though still regular pronoun paradigm.