Just as the federal government is split into three branches, state government is divided into a judicial branch, legislative branch, and executive branch. The legislative branch passes laws, the judicial branch interprets laws, and the executive branch carries out laws. At the state level, the governor represents the executive branch and is responsible for carrying out laws.
The governor's responsibilities include signing laws passed by the state legislature, serving as commander in chief of the state's military forces (if they have any), convening the state legislature for special purposes, delivering an annual State of the State address (similar to the President's State of the Union Address), providing the state legislature with a recommended annual budget for their approval, granting pardons to state prisoners, and declaring special elections to fill vacant positions in state government.
The question asks what powers you would give the governor in addition to those that they already have. This would mean taking powers that are otherwise granted to the state legislature and judiciary or the federal government and giving them to the governor. It requires you to first consider whether you think that the governor currently has enough power. If you believe that the governor should have additional powers, you then must consider what those would be.
If you feel that the governor should have greater power to interpret the law, then you would remove power that is currently vested in the judiciary and give it to the governor. If you feel that the governor should have greater law making power, you would take power from the legislature and give it to the executive.
My suggestion is to look at your specific state and see what specific powers are vested in the judiciary and legislature (specifically those belonging to the state house of representatives and state senate individually) before answering the question. At that point, you will have a better idea which specific responsibilities other branches of government have that you feel should be given to the governor.