I sometimes hear 姉さん used as just another way of addressing an older woman but I don't understand the usage of 親戚の姉さん. 親戚(shinseki) is relative so does Shinseki no Nesan mean the literal older sister of a relative, or something else? The full fragment I want to get the meaning of is 昔から大好きな親戚の姉さん.
This 'の' should most naturally be regarded as an apposition, rather than possession. So it refers to a 姉さん, who is your 親戚.
Looking up a dictionary, the definition of the word 姉さん usually starts with these two:
- older sister (広辞苑: >「あね」の軽い尊敬語…)
- young lady (広辞苑: > 若い女性を呼ぶ称。)
In the phrase 親戚の姉さん, it falls somewhat in between. I think "a female, comparative to your older sister (existent or not) in age" would be a fair interpretation. (I initially felt that the phrase typically refers to a young person, but I won't find it odd if the speaker is 55 yo and his '親戚の姉さん' is 60 yo.)
So it refers to someone in your relatives, who is a little older than you. Could be your grandfather's sister's granddaughter or anything. Note that the speaker's real sisters will not be included in the phrase.
"A relative's older sister", on the other hand, is a possible interpretation but weird without context. As you noticed, a relative's sister is your relative, so it isn't logical to mention separately. Unless that relative is known beforehand, as in this (possibly unnatural) random example I came up with.
この前親戚から仕事を受けました．その親戚の姉さんが，今回の依頼主です． An older sister of that relative (with whom we had a contract last time) is our client this time.