In "kind of you", "you" is (apparently) an "experiencer NP"
According to "Predicate-Argument Structure of English Adjectives", by Akira Ikeya, the prepositional phrase headed by of in an adjective phrase like "kind of you" expresses the "thematic dimension" of the adjective, and has the thematic role of "experiencer" (126.96.36.199).
I found another post on this site that also seems to suggest that the object of "of" in these contexts might be categorized as an "experiencer NP": John Lawler's answer to the question "How does the to infinitive work with adjectives like “wrong” and “wise”?.
The predicate adjectives (be) wrong and (be) wise are flip psychological predicates, which means they have an experiencer argument, which may be the subject, as in the first sentences above.
But the experiencer NP can also be expressed as the object of a preposition (of with these predicates) [...] This experiencer NP is coreferential here with the subject of [a following] infinitive clause.
I'm pretty mystified by the meaning of "experiencer" in this context: I would guess that it is some kind of jargon, since it doesn't make much intuitive sense to me to call the person who is being kind an "experiencer". But maybe you will be able to find some explanation if you look at some more sources that use this term.
There may be some more relevant information in "Syntax and Semantics of of in the Construction "It is A(djective) of NP to VP"-Synchronic and Diachronic Approaches", by Fuminori Matsubara (2000). Matsubara agrees with CGEL (quoted in Shoe's answer) in classifying the "of" prepositional phrase in this construction as a complement of the preceding adjective (p. 72).
we can summarize the semantic characteristics
of of and the relation between the adjective and of NP in
this construction, as in (26) and (27), respectively:
(26) The core meaning of of:
Of denotes Provenance and Inherence of something
characterized by an adjective.
(27) The relation between the adjective and of NP:
Of NP is selected by the preceding adjective as its
The idea that of denotes "Provenance and Inherence" is attributed to "Hosoe (1942: 35, fn. 1/103, fn. 1)"; the cited source is given as "Hosoe, I. (1942) An Advanced English Syntax, Taibundo, Tokyo."
There seem to be references to other relevant papers in section 2.1.1. of The Description of Adjectives for Natural Language
Processing: Theoretical and Applied Perspectives. I particularly want to get my hands on "Arnold, D., Theoretical and descriptive issues in machine Translation, Phd dissertation, University of
Essex, 1989" and "Silva, G. and S.A. Thompson, “On the syntax of adjectives with ‘it’ subject and infinitival complements
in English”, in : Studies in Language, 1:1, 1977, pp. 109-126," but I haven't yet.
Other adjectives like kind
The prepositional phrase "of you" acts as the complement of the adjective "kind" in the sentence "That was kind of you". The grammar of complements often depends on the identity of the head word, so I think that it would help to look at the grammar of the adjective kind.
There are a number of other adjectives that behave similarly to "kind", and I am familiar with some literature on this category of adjectives that I summarize in my answer to the question "Is the sentence “Queueing is so thoughtful of you.” grammatically correct?"
Based on the literature that I have been able to find so far, there seems to have been more interest in explaining how the adjective and prepositional phrase relate to a following to-infinitive (as in "It was kind of you to do that") than in explaining how the of-prepositional phrase functions.
Why I avoided using the term "genitive" in this answer
A side point: I think I would disagree about modern English having five cases. That analysis seems overly based on the grammar of other European languages. No English word has a distinct vocative form, or a dative form that is distinct from the accusative form. Furthermore, the functions that are carried out by the "genitive" form in other languages are divided among several distinct constructions in English: e.g. our, ours, of us, of ours. So I would call "of you" a prepositional phrase (headed by the preposition of) and avoid calling it any kind of genitive.
I don't think this really affects the substance of your question much, if at all: I have just treated it as equivalent to the question "what kind of prepositional phrase is 'of you'?"