If you expect her to actually read the rulebook and learn the game more-or-less on her own, then the 1983 red-box D&D Basic Set (edited by Frank Mentzer, with cover art by Larry Elmore) is by far the best choice. It's at a reading and content level suitable to 9-10 year olds (but not totally insulting to older readers like the later Black Box reputedly was), and after reading through the first book (and playing the choose-your-own-adventure style intro adventures) even a total newbie with no previous knowledge or experience whatsoever should have a very good grasp of both the principles and the rules of the game. By reading through the second book that same newbie should be able to DM the game as well.
If, however, she's going to be playing under the guidance and tutelage of more experienced players (i.e. you) then any of OD&D (1974), Holmes Basic (1977-78), Moldvay Basic (1981), Mentzer Basic (1983), or the Black Box (1991) should all work equally well -- the rules are mostly the same between them, and much simpler than either edition of AD&D or d20 (not as many numbers, modifiers, special cases, etc.).
If, however, she's going to be playing under the guidance and tutelage of more experienced players (i.e. you) then any of OD&D (1974), Holmes Basic (1977-78), Moldvay Basic (1981), Mentzer Basic (1983), or the Black Box (1991) should all work equally well -- the rules are mostly the same between them, and much simpler than either edition of AD&D or d20 (not as many numbers, modifiers, special cases, etc.).
