The correct answer is H.
The governor may appoint a judge to fill a vacancy in a state court when a position becomes vacant before the end of the term, such as following a judge's death or resignation.
Texas generally uses elections to select its state judges, but the governor has the power to make appointments to fill certain judicial vacancies until the position can be filled as provided by law.
The other choices are incorrect. In particular, the Texas governor does not appoint federal district judges. Federal judges are nominated by the President of the United States and confirmed by the U.S. Senate.
Therefore, H is the best answer.