Hello jennybro and WELCOME to the forums!
You sure picked a dandy topic! The correct use of pronouns is important, and at times very confusing. At first glance, this is how I analyzed the sentences:
1. The newest member of the team was (her, she).
The verb "was" is a linking verb that is connecting the girl/woman to the subject, "member". Therefore we are dealing with a predicate nominative, in this case a pronoun. The nominative case of the pronoun is she. So, the answer looks like this:
=> The newest member of the team was she.
Invert it and it still makes sense: She was the newest member of the team. Actually, to my ears, it sounds much better this way!
2. The medical-research grant winner was (he, him).
Once again, there is the linking verb "was" connecting the guy to the subject, "winner". It's a predicate nominative: he.
=> The medical-research grant winner was he.
List of Linking Verbs:
to be to become to grow to turn to prove
to look to feel to sound to smell to taste
to appear to remain to keep to stay
You can tell the function of a verb, whether it's linking or describing an action, by putting what follows the verb next to/in front of the subject. If it makes sense, then the verb is a linking verb. If not, then the verb is an action verb.
1. My mother's cakes look gorgeous. (gorgeous cakes= linking verb)
My mother looks at me lovingly. (lovingly mother = action verb)
(You cannot have a lovingly mother = lovingly is an adverb)
2. My mother feels old. (old mother = linking verb)
My mother felt my arm to see if it were broken. (my arm mother = action verb)
Linking verbs use the nominative case. So, that is why I put all of this here! I sincerely hope this helps you to be able to decode future similar sentences.
Best of luck to you!