Mrs Thatcher had been among those cheering the future foreign secretary when, as a 16-year-old schoolboy, he delivered a barnstorming speech which took the Conservative Party conference by storm.
However, she was rather less impressed when - as a 21-year-old Oxford graduate - he tried to secure a prestigious posting as special adviser to the Chancellor of the Exchequer.
Papers released by the National Archives at Kew, west London, show she angrily blocked the move, denouncing it as a "gimmick" and an "embarrassment" to her government.
No 10 received the request for her approval for Mr Hague to act as an adviser to the Chancellor Sir Geoffrey Howe and Chief Secretary Leon Brittan in a letter from John Kerr, a senior Treasury official, dated March 17 1983.
"The Prime Minister will I am sure remember his 1977 Party Conference speech as a 16 year old schoolboy!" said Mr Kerr, no doubt hoping to gain her approval. The effect, however, was entirely the opposite.
She scrawled across the top of the letter in thick black ink : "No (triple underlined) - this is a gimmick and would be deeply resented by many who have financial-economic experience."
Instead, she endorsed the assessment of her private secretary Robin Butler who noted: "Promising though William Hague is, it is a bit difficult to see what a 21-year-old will contribute as a special adviser in the Treasury."
In his formal reply to Mr Kerr, Mr Butler wrote: "The Prime Minister said that, however promising Mr Hague is, the appointment of someone so young and with so little experience would be an embarrassment to the Government and would be resented by more experienced people in the Conservative Research Department.
"She also had reservations about the degree of access to Treasury policy papers which appointment as a Special Adviser would give Mr Hague.
"She suggested that, if the Chancellor and the Chief Secretary wanted Mr Hague to help with speeches, he should be employed by the Conservative Research Department or some other private sources."
But with news having already leaked out that Mr Hague was going to work at the Treasury, Mrs Thatcher's press secretary Bernard Ingham had to brief journalists that the reports were "incorrect" and that he had been given a much lowlier posting at the Conservative Research Department (CRD) on the "economic side".
A source close to Mr Hague made clear he bore no hard feelings, saying his time at CRD had been "a wonderful introduction to politics at a high level".
"The Foreign Secretary thinks that Margaret Thatcher was, as usual, right," the source said. "He is still very proud that Margaret Thatcher gave him her backing when he stood for the leadership of the Conservative Party 14 years later."
One future government minister who fared rather better when it came to getting a first foot on the political ladder was Oliver Letwin.
Mrs Thatcher had no reservations at all when Education Secretary Sir Keith Joseph requested permission to make him his part-time, unpaid special adviser while working at the CRD.
Unlike Mr Hague, however, Mr Letwin - who is now the Conservative policy chief - enjoyed the advantage of family connections.
In his letter to the Prime Minister, Sir Keith wrote: "You know his parents, Shirley and Bill Letwin: Oliver, after a brilliant career at Cambridge, has just come back from a year at Princeton. He is a particularly mature as well as intelligent young man."