Give Them Two Hats
In a federation, the central government and the member states share power, each claiming authority over different domains. Over time, however, the central government tends to annex more and more functions to itself. Power that was once distributed becomes consolidated. This destroys the purpose of having a federal form of government (diversity, competition, experimentation).
The following idea may sound strange, at first. Allowing the governor of a state to double as that state's senator might help to prevent the centralization of authority.
State governors hold the greatest amount of political power within their local jurisdiction. If they savor personal power, as most politicians do, then they are loathe to surrender any authority to the central government, where they have no power. This can be seen today; liberal and conservative governors alike have been known to bemoan the encroachment of the federal government upon their personal kingdoms.
Senators, on the other hand, have no other position of power; thus, they generally vote to increase the authority of the central government at every opportunity so that they might have more. Nothing discourages them from doing so.
By forcing the governor of a state to act also as the senator of the state, we might use politicians' own avarice to preserve the sovereignty of individual states, thus allowing for more diversity and experimentation throughout a decentralized republic. When forced to choose, it seems probable that Senator-Governors will do their best to keep the power at the state level, where they are the prime movers, rather than surrender it to the federal level, where they are but one ruler among many.
In addition, state governors are kept busy by local administrative tasks, which means that Senator-Governors would not have as much time and energy to dedicate to the court intrigue of federal politics. For preventing the growth of the central government and protecting the rights of individual citizens, this is a good thing.
The realistic benefits of adopting such a proposal in the America of today are debatable. The US Senate, after all, requires two senators from each state, not one, making this policy only half as effective as it might be.
The presence of the House of Representatives, a body dedicated to national majoritarianism and the alternating domination of one faction over another, makes the proposal all but pointless under the existing system.