What is the grammatical role of *your* in "... by your being ..." phrases?

Solution 1:

The subordinate clause

• your being well cared for

is the object of the preposition by.
That makes it a noun clause, i.e, a clause that acts as a noun.

Here's an elaboration of what I said about noun clauses in this answer:

Noun clauses (also called Complements) act like nouns, as subject or object. The noun subordinate clauses are boldfaced in the examples below. There are four varieties, each with its own complementizer. Two types are tensed; they must have a subject and a tensed verb. Untensed (non-finite) clause types -- infinitives and gerunds -- often have missing subjects.

Subject: For me to leave early would be a mistake. (infinitive "for-to" complementizer)
Subject: Leaving early is not recommended. (gerund complementizer, missing subject)

Subject: What he told me is not for publication. (embedded question tensed complementizer) Subject: That I have to leave early is unfortunate. (that tensed complementizer)

Object: They told me to leave early. (infinitive complementizer, missing subject)
Object: I hate his playing the piano at all hours. (gerund "POSS-ing" complementizer)

Object: I didn't hear what she told you. (embedded question tensed complementizer)
Object: They told me that I had to leave early. (that tensed complementizer)

Other names for the infinitive complementizer include "for-to complementizer",
because it has two parts:

1. to marks the verb phrase of the infinitive clause, while
2. for marks the subject, obligatorily in the infinitive subject clause beginning the sentence.
   For her to win was not unexpected. ~~~ *Her to win was not unexpected.

Everybody knows the to part; omission of to in infinitives like do in let me do it is unusual.
But the subject is more often deleted than present in infinitive clauses, and even when it is present, it is often seen without for. That's infinitives.

The other untensed type is the gerund clause, and once again there are two parts to the complementizer, one marking the subject and one the verb phrase.

1. Either
   POSS (Bill's, my, your, the old man's, etc.)
   Or
   ACC (Bill, me, you, the old man, etc.)
   marks the subject of a gerund, if present. I.e, you get your choice how to mark the subject --
       your being well cared for and you being well cared for are both grammatical.

2. The first verb in a gerund verb phrase must have the -ing verb form.
   The technical names are the "POSS-ing complementizer" and "the ACC-ing complementizer".

So, to answer the questions, the grammatical role of your is Subject of the gerund clause.
I would guess the dissonance comes from the fact that

• You're being well cared for.

is a grammatical and homophonous assertive main clause that is, however, tensed. So its subject can't be deleted and can't be possessive or accusative. Whereas

• your being well cared for

is not tensed, not a main clause, and is presupposed instead of asserted, so its subject can be anything except a nominative pronoun (I, we, he, she, they). That's enough to confuse anybody.