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I'm applying to a school from where an older sibling graduated a few years ago. Would it be appropriate to mention that alumni connection in my personal statement? It wouldn't be gratuitous or anything, in fact my purpose in mentioning her would be to say that, through her, I became acquainted with the school and since then have always made it a goal to try and gain admittance there. But what do you all think? Any advice would be welcome.

From my experience plus others, a lot of people only write one PS so because of page limit/flow a long personalized section probably wouldn't fit well, but if you can make it flow by all means include it. If not, it seems like a great Why Essay for whatever school you're applying to. Definitely fit it into the application especially since it has influenced your decision to apply.