Instructions given to someone to observe a situation.
A brief is a summary of the facts of a legal case drawn up for the instruction of counsel conducting the case in court - literally an abbreviated synopsis of the facts. It isn't surprising that the term 'watching brief' originated in the legal profession. Lawyers are sometimes given 'a watching brief', that is, instructions to observe a case but take no active part in it.
The expression was coined in England in the early part of the 19th century. It was used throughout the legal profession but all the early instances of it that are to be found in print come from East Anglia. An example of that comes from The Norfolk Chronicle, March 1814:
There was virtually no defence to this action, the defendant's counsel only holding what is technically called a Watching Brief, to see that the case was fairly made out.
Other early citations, from 1816 and 1820 respectively, are found in the Bury and Norwich Post and the Stamford Mercury's reports of the Lincolnshire Assizes.
In the 20th century, the use of the term has widened and is now used outside legal circles. Anyone who is given the task of observing the progress of some situation or other may be said to have a 'watching brief'.