In "The Smoke Screen", Dr. Thorne's anti-smoking proposals are viewed as impractically radical and impossible to implement, and he is eventually Kicked Upstairs into a job in the Treasury to get him out of the way. While convincing him to accept the Treasury job, Hacker — who has simply been using Thorne's proposals as leverage for some less expensive but equally unpopular proposals of his own — argues that the problem with getting Thorne's policies implemented would be with the Treasury, not the Department of Health, and frames the new job as a way of getting them implemented via the back door. Fast-forward thirty years, where Dr. Thorne's seemingly ridiculous proposals are now, in the real world, widely accepted as government policy towards smoking. Apparently Dr. Thorne was more successful in the Treasury job than anyone expected...
In one episode, Hacker has to give someone a job. Sir Humphrey asks him if the guy watches television. On being told that the guy doesn't even have one, Humphrey says "Fine, make him a governor of the BBC." This may sound like a joke about the BBC, but actually the Board of Governors of the BBC weren't part of the BBC itself. They were a government-appointed body that was supposed to oversee the BBC's operations on behalf of the taxpayers, and at the time the show was being made, Margaret Thatcher was regularly using the Governors to put pressure on the BBC to be less critical of her government and its policies. The Governors were widely resented inside the BBC and were eventually replaced by the BBC Trust... which in turn has been replaced by other, similar bodies since then.
Since it's the Department of Administrative Affairs (rather than a ministry), surely Hacker should have been a Secretary of State for all of the first three seasons.
No. And Yes. 'Minister' is a title used for both Ministers and Secretaries of State in informal discussions. The transition to all departments having a Secretary of State started in the '60s and was not complete until the '90s.
The Doylist answer is probably that Yes, Minister as a title / catchphrase is more catchy than Yes, Secretary of State. As for an in-universe answer, it's possibly just one of those little quirks of the British cabinet system (sort of like the minister in charge of the Treasury being known as the Chancellor of the Exchequer rather than, say, the Treasury Secretary).
The Watsonian answer is that the Department of Administrative Affairs is a Kicked Upstairs position by all accounts as Hacker, despite becoming Prime Minister through a combination of Xanatos Gambit and sheer dumb luck, is shown to have very little real power. He's constantly dealing with petty crises and the few real crises he deals with are usually footballs which have landed in his lap. The vagueness of the title, of course, is due to the fact Hacker can and does deal with everything the writers want him to do without any according respect.
While most departments today are headed by a Secretary of State, this wasn't really the case in the 1980s; the process to streamline things this way only began with Harold Wilson in the 1960s but, like most things with the British political system, it took its sweet time. Presumably no one had gotten around to changing the official title of the person who headed up the Department of Administrative Affairs when Hacker was put in charge.
Humphrey's Sesquipedalian Loquaciousness is often talked about as if it's merely a smokescreen, or an attempt to bamboozle Hacker, and that's how it's normally used in the show. However, the great British civil servant Sir Ernest Gowers, who at one point during World War 2 was in charge of Civil Defence for the whole of London, wrote a classic guide to plain English for civil servants, Plain Words, in which he pointed out that although civil servants needed to use clear language when talking or writing to the public, sometimes, they have to communicate with each other using legal language or technical terms that seem impenetrable to others, simply because their communications have to be absolutely unambiguous in their effect. He cited an Order from 1945 which read The Rags (Wiping Rags) (Maximum Charges) Order 1943 (as amended) shall have effect as if in Article 1 thereof for the figure '8' where it occurs in the last line there were substituted the figure '11 1/2'. Gowers noted that this doesn't mean anything to anyone except those who know the document to which it refers, and for whom it's completely unambiguousnote What it actually is, is an order allowing the people who wash wiping rags to charge more for that service than they did previously, whereas a version of it that was more self-explanatory might introduce undefined terms that could lead to ambiguity or difficulty in application.note Plain Words is so useful as a guide to writing clear official English that it's been through four editions and is still in print today. Humphrey's linguistic habits have clearly been formed by having to write in this kind of way; it's just that he weaponises it, so as to say things to Hacker in such a way that Hacker won't understand him. Bernard and Humphrey even lampshade this for Hacker in the "Rhodesia situation" joke in "The Whiskey Priest", in which they show Hacker how he can be got out of a compromising situation by bringing it to the PM's attention in the form of a reference to a piece of legislation, not by actually telling the PM what's going on: the result will be that the PM won't notice what's going on because of the way it's been phrased, but at the same time, Hacker can get out of trouble later by pointing to the fact that he did bring it to the PM's attention.