Friend of friend: She's Happy
Me: Is it because of her meeting me/having met me?
Can I say it that way rather than is this because she has met me?
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In most situations, I would use your top examples ('her meeting me' and 'having met me') long before I would use 'because she has met me'.
It depends what meaning you want to convey. Your three examples do not mean exactly the same thing. (BTW '...because of her meeting me...' is a bit awkward, probably because the 'her' is redundant).
If you say 'She is happy because she has met me' that will be understood as referring to a single meeting. Either you are someone really famous whom she had been hoping to meet or you have such an amazing personality that a single meeting makes her happy.
'...because of having met me' carries similar connotations.
Not being famous or having an amazing personality, I would nevertheless like to think that my wife of many years is happy 'because of meeting me' (and, by implication all the wonderful things that followed from that meeting).