"She remembered me."
Translation:Lei si ricordava di me.
AFAIK mi is a pronoun that can function either as direct or indirect object. In lei mi ricordava it serves as direct object. In order to operate as an indirect object and for the sentence to mean that she reminded me [of something] an explicit direct object would be needed.
'me' is used when there are two clitic objects. "lei me lo indica ogni giorno" (she shows it to me every day) replace the 'e' with 'i', and the indirect object clitic always precedes the direct object clitic. http://www.uvm.edu/~cmazzoni/3grammatica/grammatica/personalpronouns.html
I'm not a native speaker so this is simply from what I have seen in past examples, as Dev says above, this is reflexive, so it is literally saying she herself remembers of me or about me. Someone more versed in the language can correct me if I'm wrong, but I think that is the gist of it.
I think the verb "to remember" or "ricordarsi" (one remembers) must be reflexive; i.e. you must have it say "si ricordava", and because it is reflexive you cannot use a clitic for the "of me" part, so you need "di me" at the end. It would literally translate to "she remembered of me".