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She will not object to his delivering the speech as long as the speech makes sense.

I think him sounds better but my teacher said it was his, but she didn't explain why. Do you know why we should use his here and not him?

thank you!
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'Him' is increasingly common in informal speech, but it is still wrong when writing carefully. 'She' is not concerned about him; she is concerned aout his speech delivery. Compare:

I don't like him.
I don't like his dog.
I don't like his talking during class.
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What if it's female?

Would it be

I don't like her.

vs

I don't like her walking to the park.

vs

I don't like her talking during nap time.
Well, the 2 sentences are not 'vs' (opposed to each other). They both use the possessive pronoun (her/his walking, her/his talking). But yes, the female pronoun has the same form (her/her) while the others have different forms (me/my, him/his, them/their, us/our, you/your), and that may be one of the factors leading all of the object pronouns to become more common in this situation.