The Bible has traditionally been used at Presidential swearing-in ceremonies. This tradition should not be changed. Although a couple of presidents did not use the Bible, most have, and it should be required. If a President is not willing to swear in using the Bible I would question his or her integrity and motives.
There have been many instances where the wrong person has been imprisoned for a crime, even a so-called capital crime. What if one of these people was put to death for these crimes? Where is the justice in that? I think that the death penalty is leveled moreso as a result of people's interpretation and anger over the crime committed.
Though a Bible is required for many cases of swearing-in, it seems odd for one to be used in the swearing-in of a U.S. president. Corruption and unfulfilled promises are practically to be expected in most (if not all) administrations, and the use of a Bible will not change that fact so it seems rather useless to make it required.
The 1st amendment gives all citizens the freedom of religion. Forcing someone to swear on the bible spits in the face of this by forcing elected officials to express Christian religious beliefs. Even if other bible's a provided for other religions, you would force the nonreligious to express religious belief. Also, someone swearing on something they don't believe in makes the swear worthless, honestly.
Should definitely be up to the president. It depends if they believe in god or not. Some people even believe in god but don't think it interferes with day to day life. Deism is what it is called, and really, what does god have to do with being president anyway?
If they feel that their chosen holy book is a sufficiently important symbol that to swear on it increases the strength of their oath then they should feel free to do so. If they are either not religious or feel that they should not swear on a religious book in relation to an official government thus protecting a strong separation of state and religion that should also be allowed. The choice is both a personal religious and a political statement that a president should be allowed to make.
Given the stated - and actual - freedom of religion in the US, I don't think this will ever be a requirement, implied or otherwise. It does present a unique opportunity for an incoming president to express his/her beliefs. I like John Q. Adams's use of the law book.
That said, swearing on the Bible shouldn't be taken as a sign (since I'm pretty sure not all of our leaders were devout Christians). It's just a nod to tradition. I get irritated when people try to scrub every trace of religion from every official act. Plenty of uses - like this one - are harmless. They stem from a time when most Americans were Christians, and the mention of that religion wasn't something anyone thought about.
The incoming president will swear on whatever he/she wants, or nothing. Like everything the president does, it's a statement. This is the first presidential act, so it's especially fraught with meaning, even in these days. I'm curious to see how president-elect Trump handles it (WHY, America?)
The use of the Bible for swearing in the president is a tradition, but it should not be required. In fact, requiring it would be unAmerican. The Constitution promises not only religious freedom but declares the separation of church and state. To require the president to use a Bible would go against both of those. At any rate, if a president-elect did not want to use a Bible he/she would most likely not want to for religious reasons in the first place and the Bible would not hold the meaning for him/her as it would to someone who did want to use a Bible anyway. The whole purpose of swearing in with a hand on the Bible is because a Christian would take any promise made on the Bible more seriously as it is sworn to God on God's word. If the person does not hold the belief to make the use of the Bible relevant than it serves no purpose to insist on the use of the Bible to swear them into office.