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In general you get to open with a prayer from X when a congressperson or senator sponsors X (usually a constituent) to open that day. When no one wants the honor, the Chaplain gives the prayer.

One person asked a Chaplain if he looked out and prayed for the Senators. He replied. "No, I look at the Senators and pray for the Country!"

Actually, I'm fine with the general principle that each member gets a shot at sponsoring someone. That way, the vast majority of prayers come from the perspective of the vast majority, but if you have a minority who has gotten elected (or is to be honored by an elected member), you get Jews, Hindus, whatever. Fair all around.