I wouldn't feel comfortable with writing a rec for someone I didn't know too well.
If a friend's, neighbor's, or coworker's daughter or a friend of one of my (hypothetical future) children approached me and I knew them well enough and felt they'd be an asset to AEPhi, I'd write a rec without hesitation. (It goes without saying that I'd recommend any daughters I may have.
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If someone I knew less well approached me, I'd probably invite her over for coffee or something, chat with her, get to know her a little better, and then write a rec based on that.
If someone I barely knew came up to me out of the blue and said "Hey, I heard you're an AEPhi, I'm going to be rushing this fall, will you write a rec for me?" I'd decline. By writing a rec that says "I barely know this girl but I think she'd make a good sister" I am doing her a disservice. Plus, when you write a rec, you're putting your reputation on the line, and I'm not willing to do that for someone I barely know.