Learn Spanish (Latin America): El negocio de mi cuñado es grande, y el de mi yerno también. - My brother-in-law's business is big, and my son-in-law's is too.

Here, el replaces the noun negocio (business) so we don't have to say it twice. Because negocio is a masculine, singular noun, we use the masculine, singular article el. Combined with de, el de mi yerno literally translates to "the one of my son-in-law."

Could I use lo de mi yerno instead of el de mi yerno?

Not in this context. Because you are referring back to a specific masculine noun (el negocio), you must use el. The phrase lo de mi yerno uses the neuter article lo and would mean something like "the situation with my son-in-law" or "the matter concerning my son-in-law," rather than referring to his business.

How would the second half of the sentence change if we were talking about "companies" (las empresas) instead of a "business"?

You would have to change the article to match the gender and number of the new noun. Since empresas is feminine and plural, the sentence would become: Las empresas de mi cuñado son grandes, y las de mi yerno también.

Why isn't there a direct translation for the apostrophe-s ('s) used in "son-in-law's"?