add 's to the singular form of the word (even if it ends in -s):
the owner's car
James's hat (James' hat is also acceptable. For plural, proper nouns that are possessive, use an apostrophe after the 's': "The Eggleses' presentation was good." The Eggleses are a husband and wife consultant team.)
"James' hat" may be acceptable, but it is clearly wrong. James's name is James, and there is only one of him. I will grant that the intention of the apostrophe in that phrase is disambiguated by the fact that the word hat is singular. However, there would be no need to refer to that implication if the writer's use of apostrophes was consistent between singular and plural nouns.