I had this buried in comments for quite some time now, but it really deserves to be elevated to an answer.

This construction is indeed peculiar. Contrary to what the accepted answer seduces us to believe, it actually is not quite analogous to "I know who shot him" and the like. Geoff Pullum over at Language Log explains why:

It's very important here to distinguish two separate structures for who's next. One of the two is an interrogative content clause, and that's commonplace. There is nothing remarkable about examples like this:

• I wonder who's next.

• Let's go and inquire who's next.
• Who's next is completely unclear.
• Who's next doesn't matter.

In all of these, the who's next is interrogative. [...]

Now, interrogatives often have exactly the same form as corresponding fused relatives [...]: a noun phrase constituent in which, in effect, the words the thing that (or the thing which, or that which) are fused into the single word what.[...]

In general, fused relatives with who just aren't used in contemporary English. In Shakespeare's time it was commonplace (recall Iago's remark in Othello: Who steals my purse steals trash; 'twas mine, 'tis his, and has been slave to thousands). It survived down to the 19th century. But it did not survive down to the present day.

Except in this peculiar use in coffee shops and the like, because in Can I help who's next? we have a fused relative construction: it's the object of help. [...]

Things are completely different if we consider not the lexeme who but instead the compound lexeme whoever. That (like all the wh + -ever words) is freely used in fused relatives. If I was hearing Can I help whoever's next? I wouldn't have written this post at all.

Emphasis added.

So why did this construction survive in some places? And in which? Geoff Pullum initially observed it in Rockport, Massachusetts. He later adds, "Lots of people have now written to me to confirm hearing or using the expression in coffee shops, bookstores [...], up to about fifteen years ago [that is, around 1990], especially in the upper Midwest, which could be the cradle of the phrase." At the same time, M. Lynne Murphy of Separated by a Common Language fame has been hearing it "a lot in shops and cafés in Brighton". She picks the ball up from where Pullum left it and entertains some theories.

Some of you may be thinking "Ha! I knew that such ungrammaticality must be an American aberration imported into the youthspeak of Britain!" But by Pullum's account, this is not a new construction, but an old use of who that had been thought to be extinct for at least 150 years. So, what's going on here? Is it that:

1. this use of who died out in most places but survived in little pockets of AmE and BrE and may be making a comeback?
2. this use of who is a natural development in English grammar that has erupted on two continents at vaguely the same time after going out of fashion for a while?
3. the phrase can I help who's next? is an idiom that was (re)invented in one country and found its way to the other?

In the UK, I'm mostly hearing it from younger people [...] I can't imagine that that many Marks and Spencer assistants/clerks spend a lot of time in the US picking up phrases that seem to be used by a minority of AmE speakers (not necessarily in the touristy areas). [... T]hat makes me lean against hypothesis (1). I'm liking (2), but really have no empirical evidence for it.