I just began reading How to Pick a President, one of Christianity Today's new line of ebooks. (Yes, I'm reading it on my new Kindle.) The first criterion that the author discusses is virtue. The virtues specifically mentioned are moral courage, temperance, commitment to justice, commitment to keeping one's word, and prudence.
The importance of picking leaders who exhibit these virtues is that we can trust them to make hard decisions in the reality of political life. If a candidate says that he (or she) will do x, y, and z relative to immigration, the reality is that by the time that Congress has had its say, and that various realities have been dealt with, x, y, and z may have changed into m, n, and o!
Instead, we should pick a leader who tells us what justice toward illegal immigrants who are already here looks like. And what it means to her to protect our borders. And how we will guard the rights of citizens as opposed to those who are here legally or illegally. And so on. Then we can decide whether or not we agree with those values. And then we decide whether these values are in fact just, and whether the candidate has had a record of keeping his word and exhibited the moral courage to stand up for what he believes to be right even if it costs him power, money, or prestige. And if that person is elected, then we trust her or him.
The book I mentioned above mentions that it is important that we look to a candidate's commitment to these values in their public and (hopefully) also in their private lives. This means that what a candidate said and did before and after they entered public life is very important. How else can we measure their fitness for leading us? Those who wish to lead us but who keep their private lives hidden perhaps shouldn't be trusted.