Not sure what my preference is here. I know that in Connecticut, the party candidate selects his/her running mate, and they are elected on the same ticket with the gubernatorial candidate.
I think it should work like that. Lt. Gov has the same dynamic as VP on a state as opposed to national level. He/she should be selected by the Gubernatorial nominee and the two should be elected together (as a ticket).
My opinion is as such because in a close election, the Lt. Gov and Gov. can be of different parties, if they're elected separately. Situation in NC currently, if Roy couldn't serve, the (Acting) Gov's party would be different from the elected Gov's party. Dan Forest is Lt. Gov (R-NC). Regardless of party, it's unfair to the voters to allow a party 'coup' like that.