Wilson and Diana home, which they purchased together in 2010, has doubled in value during the interceding years.

Correct sentence: Wilson and Diana’s home, which they purchased together in 2010, has doubled in value during the interceding years.

In this case, we know that Wilson and Diana purchased their home together. The punctuation "Wilson and Diana’s" reflects the joint ownership.

Jacksons preference for green was apparent, as everything he owned was one shade of green or another.

Correct sentence: Jackson’s preference for green was apparent, as everything he owned was one shade of green or another.

The sentence suggests that there is only one Jackson (as indicated by “he”). We know that an apostrophe is needed to indicate that we are talking about his preference. To do so, an apostrophe should be placed between the “n” and the “s” in “Jackson’s.” No apostrophe is needed in “preference,” as it does not show possession.

Rachael was able to prove to her sister, Amanda, that her’s was the right way to cook the soup.

Correct sentence: Rachael was able to prove to her sister, Amanda, that hers was the right way to cook the soup.

The error is in “her’s.” Hers is a possessive pronoun, and possessive pronouns do not take apostrophes. Apostrophes are used for contractions (e.g., she’s = she is) or possessive nouns (e.g., Rachael’s recipe), not for possessive pronouns.