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It depends how important it was not to be found out. If I was playing a character in a role-playing game, I think I'd choose something that was appropriate to the character (Mary Jane Conway for a 1960s housewife, Lady Augusta Marlborough for a 19th-century British high-society dame, etc.)
If it was just for the fun of being someone else, like writing romances, I'd just choose something I liked and that sounded good. (And that wasn't some other romance author's name.)

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If I was really concerned about being found out, I'd choose a not very distinctive or memorable name. And nothing related to my own name. Something age-appropriate; not Madison at my age, and not Edna either.

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Before you criticize a man, walk a mile in his shoes. That way, when you criticize him, you're a mile away and you have his shoes!
Steve Martin