It is said that he dedicated it to Therese Malfatti, who was his piano pupil in 1809/10 and with whom he may have been in love, though the topic is still debated. Most researchers think he was, but some disagree. Anyway, if it was for Therese, why then 'Fur Elise'? Maybe it was an error, made by researcher Nohl. Years after Beethoven's death he stumbled on the autograph of the little piece when he visited the Malfatti family. However, not one member of the family said something about a possbile 'special relationship' between Beethoven and Therese. Did Nohl misread Beethoven's hieroglyphs? Later on researchers learned that Beethoven may have been in love with Therese and they concluded that Nohl must have erred. That is what most researchers think/thought. Unfortunately the autograph is lost. Recently Kopitz published an article on the internet in which he speculated that the girl behind the bagatel is not Therese Malfatti, but Elisabeth Rockel, a singer who married Beethoven's friend and competitor Hummel. I find his evidence not very convincing. Lorenz published a strong refutation.
See:
Leitzmann, Albert. Beethoven und Therese Malfatti. Eine kritische Studie. In: Deutsche Rundschau (1911).
Brandenburg, Sieghard. Der Freundeskreis der Familie Malfatti in Wien. Gezeichnet von Ludwig Schnorr von Carolsfeld. (Bonn, 1985).
Beahrs, Virginia. The Beethoven-Malfatti Connection Revisited. In: The Beethoven Journal (1998).
Kopitz, Klaus Martin. Beethoven, Elisabeth Rockel und das Albumblatt "Fur Elise". (Keulen, 2010).
Lorenz, Michael. Die "Enttarnte Elise". Elisabeth Rockels kurze Karriere als Beethovens "Elise". In: Bonner Beethoven-Studien (2011).