I'm curious; are you? Here's the thing about the White House: art comes, art goes, and it moves around with each president. The classic example is the Roosevelt Room, where presidents routinely have swapped images of TR for FDR and back again; Bush 43 famously had FDR removed from the room entirely. So I want to know whether it is especially significant that this figure was a silent witness to Obama's second oath.
The oath was administered in the White House Map Room. As recently as 2008 this portrait did not hang over the mantel, but to the side of it.
According to the White House Museum site, what is hanging over the mantel in the image above is "the last situation map prepared in this room for President Roosevelt, on April 3, 1945." The map hung in the same place during the Clinton years.
So, who's our guy? I confess that my ability to identify people or painters of the 18th-19th century is woefully lacking. This site wouldn't tell me.This one didn't either. This got me closer, pointing out that the Map Room contains two portraits: one of General Andrew Jackson (at whose visage TR's sons purportedly threw spitballs) and an 1804 Charles Willson Peale painting of Benjamin Latrobe, Thomas Jefferson's architect. But this source has no images of the portraits and despite Jackson's distinctive head of hair - which this guy does not have - one shouldn't make claims without all the visual evidence.
Deeply significant? Probably not. Perhaps White House photographer Pete Souza simply thought a portrait in the background would look better than a map. Perhaps Rahm and Co. wanted to insert a visual surrogate for the American people, an additional pair of eyes to witness the oath. After all, Latrobe was one of many architects for the U.S. Capitol, site of the first almost-oath.