For the record, the word that Obama used wasn't bemused but bemusement. Here is the transcript of his relevant remark from April 27, 2011:
THE PRESIDENT: As many of you have been briefed, we provided additional information today about the site of my birth. Now, this issue has been going on for two, two and a half years now. I think it started during the campaign. And I have to say that over the last two and a half years I have watched with bemusement, I've been puzzled at the degree to which this thing just kept on going. We've had every official in Hawaii, Democrat and Republican, every news outlet that has investigated this, confirm that, yes, in fact, I was born in Hawaii, August 4, 1961, in Kapiolani Hospital.
In Merriam-Webster's Eleventh Collegiate Dictionary (2003), the word bemusement appears without definition at the end of the entry for bemuse:
bemuse vt (1735) 1 : to make confused : PUZZLE, BEWILDER 2 : to occupy the attention of : DISTRACT, ABSORB 3 : to cause to have feelings of wry or tolerant amusement {seems truly bemused that people beyond his circle in Seattle would be interested in his ruminations —Ruth B. Smith} — bemusedly adv — bemusement n
It seems unlikely that Obama meant to say that the assertion that he was born in a madrassa in Kenya had been absorbing his attention over the past two and a half years, so that takes definition 2 out of contention. He used puzzled three words after he used bemusement, which certainly gives a boost to the idea that he was invoking the definition 1 sense of the word; but he almost certainly did not want listeners to interpret bemusement as a one-word synonym for confusion or bewilderment. That's why definition 3's sense of "wry or tolerant amusement" is a useful adjunct to the puzzlement sense of definition 1. For this reason, I share Alain Pannetier Φ's view that Obama used the word with the intention of drawing on both definition 1 and definition 3.
Side note: How long has bemusement implied a type of amusement?
As the Eleventh Collegiate indicates, the use of bemuse to indicate a specific subcategory of amuse arose some time after the "puzzle, bewilder" and "distract, absorb" meanings did. But when did this new sense reach the threshold of popular usage necessary to merit inclusion in the Merriam-Webster's Collegiate Dictionary series?
The answer is, perhaps surprisingly, in 2003. The Tenth Collegiate (1993), like its three immediate predecessors in 1983, 1973, and 1963, had only two definitions for bemuse:
bemuse vt (1735) 1 : to make confused : BEWILDER 2 : to occupy the attention of : ABSORB — bemusedly adj — bemusement n
It thus appears that bemusement as a form of amusement is a fairly recent innovation—one based, I suspect, on a mistaken supposition that bemuse shared an element of gaiety or humor or entertainment value with amuse. In any case, the (relatively) new meaning raises some issues of ambiguity, as in the phrase "bemused smile": Does the phrase describe a puzzled smile, an abstracted smile, or a wry smile? In many written instances, especially where a character says or does something equivocal "with a bemused smile," it's impossible for the reader to tell.