The common view is that the Scottish English adjective wee means little. Doubtless it often does; but as I slowly make a little headway in learning the ways of Standard Scottish English (and its much more inscrutable sister language, Scots [SCO], which in general I cannot even understand), I have been noticing that (in Edinburgh at least) the word wee is more commonly used in a rather different way, one that couldn't possibly be thought to convey anything about diminutive size or cuteness.
When the workman who was regrouting my kitchen floor told me he was going for his tea break, he said, "I'll just be going off for a wee cup o' tea." I'm quite sure he wanted, and got, a proper workman's mug of tea, not a delicate little bone china cup. He didn't mean that his cup would be little.
If someone who has asked you to fill out a form says "I'll get ye a wee pen," don't expect it to be any smaller than a normal standard-issue pen; it's just a friendly message of comfort.
When you check in at the desk for a dental or medical or optician's appointment they will mark you down as having arrived and then say, "If ye'd just have a wee seat over there, we'll call ye in a minute or two." The seats indicated are not liliputian but of standard size, with a sitting surface about as 1.3 times the width of an average butt (1.3 bw).
In fact the phonetician Professor Martin Ball (now at the University of Louisiana, Lafayette) clearly remembers being told by a university department's reception in Northern Ireland (where a dialect closely related to Scottish English is spoken) to "Just have a wee seat for a wee minute." It was a standard 1.3 bw seat, and standard 60-second minutes were being used for the timing.
Usages of this sort are actually the majority of instances of wee that I hear. And what this usage seems to be doing is to impart some kind of friendly and encouraging attitude about this event not being a significant setback. The break for tea won't take all morning and he'll be back on the job before long; the wait for the doctor won't be too long and they'll call you quite soon… That's the sort of thing people seem to be implying by popping a wee wee in there.
I have a hypothesis about the meaning. I think wee is developing into something rather like damn, only positive. Let me explain.
Damn has the syntax of an attributive adjective but the semantics of a scowl. When you say Somebody stole my damn guitar, you aren't describing the guitar as damned. It might be a much-treasured full-bodied Martin acoustic from the 1960s with genuine mother of pearl fretboard inlays and you might love it dearly. The irritation is at the whole event, the theft and everything surrounding it.
The adjective damn can be inserted as a modifer of any suitable noun phrase in the sentence (and I grant you that "suitable" there needs some detailed explication), but its semantic contribution is always one of speaker attitude toward the whole situation. It supplies what linguists now call a conventional implicature: a commitment to an extra message that (metaphorically speaking) comes through on a second channel, without adding anything to the factual content of what is said. (The best introduction to this topic is the work of Chris Potts; see the useful elementary introduction "Into the conventional implicature dimension" in The Philosophical Compass (PDF copy from Chris's own website here so you don't have to mess with Wiley's registration and purchase system), or the masterful but highly technical study in The Logic of Conventional Implicature, Oxford University Press, 2005).
It seems to me that wee has a similar syntactic privilege of occurrence — you can just pick a salient noun at random and stick wee on that — but the semantic contribution is just an optimistic and comforting attitudinal overtone: rather than the vague impression that the speaker is pissed at the situation, which is what damn conveys, wee supplies a vague impression that the speaker is being helpful and optimistic and that things are going to be just fine. But there is no necessary entailment that anything is little.
I've cited business examples above, uttered by people who clearly aimed to please, but my friend Warren Maguire, who grew up in Northern Ireland, tells me that wee can involve something rather less innocent: a sneaky and even menacing sort of intimacy, as in A wee bullet would sort that fellow out. I don't think the sense there implies that small-caliber ammunition would be best; wee is doing its conventional implicature job, suggesting some sort of intimacy and it'll-be-all-right feeling that is rather hard to specify precisely.
Jim McCloskey tells me that in Northern Ireland you will typically be asked to provide "a wee signature" if you have to sign something in a bank or shop, and they don't mean a miniaturized one. And when they offer you "a wee receipt", that too will be standard receipt size — there's usually just the one size for them.
I'll keep an eye open for further examples, and perhaps post them here as updates. I should say that tons of people — scads of them, whole armies of them — have been mailing in to suggest that diminutives in a variety of languages (especially Romance, Germanic, and Slavic) show a similar tendency; but I think most of them are missing the point, because hardly anyone has given me even a hint of an example that really establishes the disappearance of the littleness meaning, as with the wee seat. It's the emergence of this intimate, trouble-minimizing, it-won't-be-a-bother conventional implicature that primarily interests me, not further examples of diminutives from people's favorite languages.
[Ye shouldnae hold yer breath for me tae open comments, e'en if ye're a Scot, but if ye've got a wee example for me tae consider, I wouldnae object tae a wee email. Try mail2languagelog at Gmail.com; but keep in mind that we've no staff at all here at Language Log Plaza, and we've no got the time tae read all the email that comes in as it is.]