What claims about morality do laws about driving on the right side of the road make vs laws about driving on the left side of the road?
The claim on morality is that you should follow the law to ensure the safety of others. There is nothing inherently right or wrong about driving on the right or left side of the road, but there is something inherently immoral about ignoring the laws that have been laid out to ensure good order and the safety of others. Driving on the right side of the road when I live in a society that has said that we should drive on the right side of the road to ensure uniform safety for pedestrians and drivers is a demonstration of love for my neighbor because I value his/her safety. That is a moral claim.
To add to my initial answer though, there are two answers, the theological one, and the pragmatic one.
The theological answer is that all government is instituted by God to govern the creation and implement God’s will on earth. In that sense the purpose of all laws should be to ensure that God’s will is done on earth.
The pragmatic answer is that all ethics are surrounded by a community. In other words, in the practical sense the community determines what ethical rules should be in place. In a pluralistic secular society, not all people have the same view of who God is, if there is a God, and what God’s will is for man. So from this standpoint, the Christian is to implement just laws that enforce morality that is keeping with God’s will, where and when we can. When the greater society is in disagreement with God’s will and we are in the minority, well then we will have to implement what we can and teach our kids to live a counter-cultural lifestyle in harmony with God’s will.