。
She was thrilled with my present.
Literal
[topic-は] I [genitive-の] gift [object-を] greatly was-pleased.
喜ぶ ('to be pleased, rejoice') takes を for the source of the joy in this construction — note that this is unusual: most emotion verbs in Japanese take に or で. 大変 functions here as an adverb meaning 'very much, greatly,' a slightly bookish intensifier (in casual speech すごく would be more natural). おくりもの is written in kana but maps to 贈り物 ('a gift') — softening the kanji often gives a sentence a warmer, more conversational feel. Gift-giving is socially significant in Japan, with established rituals around return gifts (お返し), seasonal exchanges (お中元, お歳暮), and souvenirs (お土産).