Learn Italian: Spero che lei gli dia il copione per suggerire la battuta. - I hope she gives him the script to prompt the line.

Because the sentence starts with Spero che (I hope that), which expresses a hope or wish. This requires the next verb to be in the subjunctive mood. dia is the present subjunctive form of the highly irregular verb dare (to give).

Are all the singular forms of dare the same in the present subjunctive?

Yes! The io, tu, lui, and lei forms of dare are all exactly the same: dia. Because of this, it is very common to include the subject pronoun (like lei in this sentence) so we know exactly who is doing the action without any confusion.

Where does the indirect object pronoun gli go when the verb is in the subjunctive?

Just like in the regular present tense, object pronouns like gli (to him) go immediately before the conjugated verb. The subjunctive mood doesn't change this rule, so it remains gli dia.

Why is per used before suggerire?

In Italian, when you want to express the purpose of an action (meaning "in order to"), you use per followed by the infinitive of the verb. So per suggerire translates as "in order to prompt" or simply "to prompt."