Chief Mouser to the Cabinet Office is the title given to the official resident cat of the Prime Minister of the United Kingdom at 10 Downing Street, London.

The noun mouser, which denotes an animal that catches mice, is first recorded in the English-Latin dictionary Promptorium parvulorum sive clericorum (Storehouse for Children or Clerics – circa 1440)—as published in 1853 by the Camden Society (London):

Mowsare, as a catte. Musceps.

The noun mouser then occurs in The points of Huswiferie, in Fiue hundreth points of good husbandry vnited to as many of good huswiferie (London: Richard Tottell, 1573), by the English poet and writer on agriculture Thomas Tusser (c1524-1580):

                    In dayrie no Cat,
                    Lay bane for a Rat.
49 Though Cat (a good mouser) doth well in a house,
yet euer in dayry, haue trap for a mouse.

According to GOV.UK, the title Chief Mouser to the Cabinet Office was first given to Larry:

Larry, Chief Mouser to the Cabinet Office
Larry has been in residence since 15 February 2011, he is the first cat at Number 10 to be bestowed with the official title Chief Mouser.

Larry, Chief Mouser to the Cabinet Office—photograph GOV.UK:

 

However, the earliest occurrence of Chief Mouser to the Cabinet Office that I have found is from Downing Street feline survives transition of power: Chief Mouser can keep post, new first lady says, published in the Ottawa Citizen (Ottawa, Ontario) of Saturday 10^th May 1997; this article by Caroline Davies was originally published in The Daily Telegraph (London, England)—Tony Blair (born 1953) became the Prime Minister of the United Kingdom on Friday 2^nd May 1997:

London—New Labour bowed to pressure yesterday and promised to safeguard the future of a long-standing member of the Cabinet Office—Humphrey, the Downing Street cat.
In a conciliatory gesture, First Lady Cherie Blair, the prime minister’s wife, posed with the tom, putting the lie to reports that she found him dirty and that he made her sneeze.
Uncertainty over Humphrey’s future had prompted a storm of protest after it was reported that the Blairs would evict him from No. 10—his home for the past eight years.
“We have been inundated with calls and letters following mistaken reports that Cherie Blair does not want Humphrey in Downing Street,” said a No. 10 spokesman.
“This is nonsense, as is the report that she is allergic to cats. Cherie and her sister had both a cat and a dog when they were growing up, and one of the first things the children wanted to see when they moved into No. 11 on Monday was Humphrey.
“This is Humphrey’s home, and as far as the Blairs are concerned, it will remain his home.”
Humphrey, a stray taken in by the cabinet office staff in 1989, became a family pet to John and Norma Major—even featuring on the official cabinet office Christmas card last year.
Like his predecessor Wilberforce, resident for 14 years until his demise in 1988, Humphrey wanders freely between Nos. 10 and 11. He is expected to resume his position on the floor during cabinet meetings, where he will be known simply as Humphrey.
Outside, he will be referred to by his official title: Chief Mouser to the Cabinet Office.
Named after the character Sir Humphrey Appleby in the television comedy Yes, Prime Minister, he became a household name after disappearing for three months in 1995. It is believed he was inadvertently posted to the Royal Army Medical College half-a-mile away as he snoozed in a Government mail van.
Tony Blair joked to the Dutch premier Wim Kok that keeping Humphrey was the “biggest decision” he had made as prime minister. Conscious that there has been a Treasury or Downing Street cat since the days of Henry VIII, he persuaded his wife to do the decent thing.

In an article published in several U.S. newspapers on Wednesday 16^th February 2011, for example in The Record (Hackensack, New Jersey), David Stringer, of the Associated Press, reported on Larry’s arrival at Downing Street in 2011, and wrote that Chief Mouser to the Cabinet Office is “an unofficial […] post”—David Cameron (born 1966) became the Prime Minister of the United Kingdom on Tuesday 11^th May 2010:

London—Prime Minister David Cameron turned to a new recruit Tuesday to help fend off his foes—an ally armed with whiskers, claws and sharp teeth.
Larry, a 4-year-old tabby cat, arrived at his new home at 10 Downing Street, tasked with warding off a pack of rats seen scuttling close to the British leader’s official residence. The former stray, adopted from London’s Battersea Dogs and Cats Home, is the latest pet to be appointed Chief Mouser to the Cabinet Office, an unofficial pest control post.
Television cameras had captured several sightings of rats around Downing Street—a common problem in London, where the trash on sidewalks provides a constant supply of food.
Cameron’s office said the pet shelter believed Larry would be well suited to his new role because “before coming to Battersea, he was a stray, so he is used to fending for himself on the streets.”
The last holder of Larry’s office, a male cat called Humphrey, retired in 1997 and died in 2006.
Ex-Treasury chief Alistair Darling—who lived next door to then-Prime Minister Gordon Brown at No. 11 Downing Street—also had a pet cat, Sybil, who was renowned for her prowess in catching mice.
Cameron’s spokesman Steve Field said Larry had taken a relaxed approach so far to his duties. “He’s spent most of the time asleep; he seems quite relaxed and easy going,” he said.