A TV program says,
they started this accounting gimmick, if you will, and they...
What does "if you will" mean? Is it a short form of "if you will [a certain verb]"?
No, it is not a short form of anything. Here, will is not an auxiliary verb, but a full verb. Nothing is omitted in the sentence. Will, here, is used in the meaning "want" or "wish", which is considered archaic in most other contexts, outside of set phrases. It is related to the German wollen, Dutch willen, etc., all with the same meaning "to want, desire".
1. (obsolete) To wish, desire. [9th-19th c.]
2. So to speak
Here is a related Language Log post that makes a rather interesting point of "if you will" being semantically strikingly similar to the ubiquitous Valley-Girl like.
Like functions in younger speakers' English as something perfectly ordinary: a way to signal hedging about vocabulary choice — a momentary uncertainty about whether the adjacent expression is exactly the right form of words or not. If the English language didn't implode when if you will took on this kind of role among the baby boomers, it will survive having like take on an extremely similar role for their kids.
Emphasis mine.
Edit: here's yet another Language Log post that actually discusses a comic strip that unfairly criticises "if you will" precisely because the cartoonist mistakes will for an auxilliary verb.
Language Log goes on to comment,
In his latest intrusion, it's refreshing to see Lemont [Brown, the protagonist] identify himself as a member of the Idiom Police, since his too-literal reading of "if you will" has nothing to do with grammar. [...] What "if you will" does not do, of course, is require the interlocutor to "accept the words you just chose to say," as Lemont claims. I suspect Lemont's the type of non-Gricean guy who responds to the discursive filler "you know" with "No, I don't know."
It then elaborates on that, linking to the aforementioned like post and quoting the definition from UsingEnglish as well as the one from OED:
Oxford English Dictionary says ["if you will"] is "sometimes used parenthetically to qualify a word or phrase: = ‘if you wish it to be so called’, ‘if you choose or prefer to call it so.’" Citations are given for the exact phrase back to the 16th century, with similar elliptical uses dating back to Old English.
It means that the speaker acknowledges that his use of the word gimmick in this case could be challenged and that he is asking for the listener's indulgence to let him call it that. So yes, it is short for "if you will indulge me."
Yes,
A TV program says, "They started this accounting gimmick, if you will, and they ....
could almost be written:
A TV program says, "They started this 'accounting gimmick', and they ....
However, the pragmatic marker if you will serves two purposes - it not only shows that the term accounting gimmick is recognised by the TV narrator to be at least slightly subjective (as do the scare quotes), but also is an interpersonal 'conversation facilitating' marker (cf 'wouldn't you agree?').
It does sound a little dated.
Will here means want, desire, wish and the idiom if you will means "If you want". In this example, it's "if you want to call it that".
It's a hedge -- the speaker is acknowledging that this particular choice of words is not necessarily the most appropriate.
Similar hedges include as it were:
they started this accounting gimmick, as it were, and they...
and, informally, like:
they started this, like, accounting gimmick, and they...
It might be easier to compare this accounting gimmick, if you like, and they... It's slightly more jarring, perhaps, but the grammar and meaning are the same.
The answers equating will with want, *wish* and desire are correct, but say what you will, the most prevalent contemporary usage really seems to be more of a hemming and hawing, that is, using filler words while one figures out what actually to say. In the initial example, dropping the phrase wouldn't really change the meaning of the sentence, so I'd drop it. But poor usage doesn't gainsay authentic usage. The similar What you will that I used above is Shakespeare's alternate title for the play Twelfth Night. And Will didn't leave anything hanging with this optional title: it was and is a standard English phrase that also afforded him a nice pun on his name, which is what Will wanted.