Yes, names should always be capitalized.
Consider: "Sam Houston was an honest man".
Yes. It should be Houston Zoo.
Houston is named after Sam Houston
Sam Houston He was a US Senator at the time
Houston was nicknamed after Sam Houston.
William Rogers Houston was Sam Houston's son.
He was not.
Sam Houston's mother was Elizabeth Paxton Houston.
Sam Houston's father was Major Samuel Houston.
Sam Houston's mother was Elizabeth Paxton Houston.
Sam Houston
Dodgeball is capitalized if it starts the sentence. But if dodgeball is used in the middle of a sentence it is not capitalized.