Im the sentence, the pen you gave her in the morning is lost"you gave her in the morning is an_?

"you gave her in the morning" is an adjective clause.

It modifies or specifies which pen.

Explanation: If the word "that" had been included, as in

"that you gave her in the morning"  it would be a relative clause; 'that' is a relative pronoun.

(In your question the "an___" is a hint that the answer starts with a vowel)

In any case, they are clauses, not phrases, because they include a subject, 'you', and a predicate, 'gave her in the morning.'