If I would like to ask a cousin who lives out of state to be the executor of my Will, could this be a problem?

Wouldn't necessarily be a problem, but realistically makes the process of the probate more onerous to carry out and conclude; advise you make a different person selection closer to you geographically.

It should not be a problem, however, the fact that the executor lives out of state could make the process of estate administration more difficult.