The elderly woman gave it to her as a present and said she really wanted her to accept it.
Literal
Elderly-woman [topic-は] it [object-を] her [to-に] presented [and-し], by-all-means accept want [quotative-と] said.
~てほしい ('I want someone to do X') expresses the speaker's desire for another's action. 是非 (ぜひ, 'by all means') adds insistence. と言った packages the desire as reported speech.