Learn Spanish (Latin America): Ella, por su parte, prefiere esperar.

While phrases like en cuanto a or respecto a introduce a new topic (meaning "as for..."), por su parte is used specifically to contrast people's actions or opinions. It translates best to "for her part," "in turn," or "on the other hand." It tells us that someone else just made a choice, and now we are seeing her contrasting reaction.

Would it be natural to drop the pronoun Ella and just say Por su parte, prefiere esperar?

It is much better to keep it. Because por su parte implies a contrast between what different people are doing, Spanish heavily favors using the explicit subject pronoun (ella) to emphasize exactly who is taking this contrasting stance.

Does the su in por su parte change depending on who we are talking about?

Yes, the possessive adjective must match the subject. If you were talking about yourself, you would change both the pronoun and the possessive: Yo, por mi parte, prefiero esperar ("I, for my part, prefer to wait").

Could I use en cuanto a here instead to mean the same thing?

You could, but the sentence structure would change. You would say En cuanto a ella, prefiere esperar ("As for her, she prefers to wait"). Using Ella, por su parte... is a very natural alternative that lets you keep the subject at the front of the sentence while still shifting the focus.