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С памятью у тебя стало хуже.
I assume this translates literally as:
It (omitted, impersonal subject) has become worse with the memory of yours.
I wonder why the noun "память" cannot be the subject instead with the meaning of:
Your memory has got worse.
Стать isn't really important here. This is just one particular modality of a catch-all Russian construction that talks about "trouble with" something or other. Which has two fairly self-contained elements, a predicate specifying how bad the trouble is (here, стало хуже), and the "c + noun" part where the noun identifies the source of the trouble — in this case, память.
The "how bad" part is very, very flexible.
E. g.
В городе неблагополучно с преступностью.
Да у него с головой плохо. [~"That one's crazy in the head".]
С котами была сплошная беда. [~"The cats were no end of trouble."]
Note that a form of быть appears in the last example — well, really, it's been there all along, just not apparently because of present-tense copula dropping.
What I'm trying to say here is, it's essentially just an accident that you can also say память у тебя стала хуже. Because you definitely can't say *в городе неблагополучна преступность or *у него плохая голова. It's just that in your example, there are two elaborations on the basic construction: the comparative хуже and the modality introduced by using стать instead of быть. Which, understandably, makes you wonder why go through all the trouble of making it impersonal and oblique when you could just make it a plain subject+verb. Whereas in fact this is a syntactic set piece where the impersonality and obliqueness are there to begin with. And it doesn't always convert this easily into subject+verb.
Admittedly there's a very small difference between saying "your memory's gotten worse" and "this memory thing of yours has gotten worse", but that difference is there.
I wonder why the noun "память" cannot be the subject instead
First of all, it can. Твоя память стала хуже is a perfectly grammatical, albeit just a little bit peculiar way to put this in Russian.
That said, all languages work differently.
In Russian we say идёт дождь (literally, "the rain goes") while in English (and lots of other languages) "to rain" is an avalent verb (a verb without object or subject). Russian simply does not have a single verb which would directly correspond to "to rain", and Russian speaking people get along just fine without it.
In the same vein Russians say "there is something at me" when English speaking people would just say "I have". There is a construct for "I have" in Russian, however its use is quite limited.
In some languages "to sneeze" and "to cough" are transitive verbs (you don't sneeze or cough, instead it sneezes or coughs you); in others you can't just "eat" or "drink", you have to eat or drink something, etc.
This is just how those languages work.
You can say:
Твоя память стала хуже.
Then the meaning will be quite the same but the word "память" will be the subject.
The noun "память" can be a subject.
Память у тебя стала хуже. Твоя память стала хуже.
It expresses a fact.
Other variants below.
С памятью у тебя стало хуже.
The same as "У тебя с памятью стало хуже", but somebody wants to accent "память".
Что-то с памятью у тебя стало хуже.
Accents unpleasant surprise, discontent.
