Learn Finnish: Hän tekee työtä perheen hyväksi. - He works for the good of the family.

Yes! While hyväksi is literally the translative case of the adjective hyvä (good), it functions here as a postposition meaning 'for the benefit of'. Just like puolesta and sijasta, it requires the noun before it to be in the genitive case, which is why we see perheen.

How does perheen hyväksi differ in meaning from perheen puolesta?

Hyväksi specifically means doing something to improve someone's situation or for their benefit (e.g., working to provide for the family). Puolesta means 'on behalf of' or 'in the name of' someone else, as if you are substituting for them (e.g., signing a document on behalf of the family).

Why is työtä in the partitive case here?

The expression tehdä työtä (to do work) uses the partitive because the work is an ongoing, continuous activity rather than a single completed task. In this context, työ functions as an uncountable mass noun.

Could I use the verb työskennellä instead of the phrase tehdä työtä?

Absolutely. You could say Hän työskentelee perheen hyväksi and it would be perfectly natural and mean the exact same thing. Tekee työtä just puts a slightly stronger emphasis on the concrete action of 'doing work'.