This Prager article has been posted a number of times. It was factually inaccurate the first three times it was posted, and is still factually inaccurate.
There is absolutely no Constitutional requirement for House members or even the President to swear on the Bible when taking his oath of office - while Presidents have traditionally done so, it's not clear that any oath needs to be with one hand on the Bible to qualify as an oath (it likely doesn't), and under the Constitution, office holders can take an affirmation of office of office instead of an oath anyway. In fact, President Teddy Roosevelt didn't swear on the Bible when he became President after McKinley's death in 1901. One simply might not have been available at the time. He still became President. And Franklin Pierce (and according to some historians, Herbert Hoover) affirmed his oath of office instead of swearing it. President Pierce's faith was shaken after he and his wife were involved in a fatal train accident - they saw their own son die before their eyes. Some sources claim Pierce made his affirmation with his hand on a law book, not the Bible.
House members are traditionally sworn in en masse by the Speaker on the first day of Congress immediately after the Speaker of the House is elected and sworn in. The 2005 swearing in ceremony is available on C-SPAN's website
here. The Speaker is sworn in around 2:13:30 by the Dean of the House; the rest of Congress is sworn in shortly thereafter.
All Speaker Hastert asked members to do was raise their right hands while being sworn in. As a practical matter, the House floor normally seats 448 (they somehow squeeze in more seats for the State of the Union address), and there are up to 439 other members of the House (including non-voting members from the territories and the Resident Commissioner of Puerto Rico) that need to be sworn in at that time. There isn't that much room on the floor for aides or family members to hold the Bible for Congressmen to swear on. So, as you can see from the video, most Congressmen appear to raise their right hand and put their left hand on nothing, at least during this ceremonial swearing in.
Now, there may be a chance for members to have a ceremonial one-on-one swearing in for photo-op purposes (or if the Member is not present at the opening of Congress or is later elected). For example, Congressman Rothman (D-NJ) has a picture of him being "sworn in" with his hand on what appears to be the Bible on his House
website. This is when Ellison might swear on the Koran - for a photo-op.
The fuss is much ado about nothing. All that's required is either an agreement to swear an oath or affirm it. No person is required to have his hand on any book. He could affirm or swear his oath on a Marvel comic book or a copy of Gone With the Wind or Green Eggs and Ham. Stop the run away irrationality, please.