Worth noting at the outset that by the 1980s, the idea of the 'British Empire' was long gone — the Falklands' status is that of a British Overseas Territory.

Edit: per /u/Xaethon's answer below, in 1982 the Falklands were classed as a British Dependent Territory.

You're right that the islands have little ostensible economic or strategic value — though it does have valuable fisheries, and there are suggestions that there may also be oil and gas reserves (which could well reignite/escalate the sovereignty dispute.)

What they do have, however, is a couple of thousand citizens who identify very strongly as British, and who hold British citizenship. Received political wisdom and public opinion in Britain dictates that they should be protected by the British government — that explains in large part why the Thatcher government went to war.

The history of the Falklands' disputed status is long and incredibly complex — there were settlements on and competing claims to the islands throughout the 17th, 18th and 19th centuries. The Argentine view is that, as one of the successor state to the Spanish empire, the islands fall under their jurisdiction (by virtue of geographical proximity.)

At particular issue is the fact that in 1833, a British naval expedition expelled the small Argentine garrison on the islands — the British, of course, see this as a simple reassertion of previously lapsed British sovereignty over the islands. The Argentines view British control over the islands as colonialism backed up by gunboat diplomacy.

The legal arguments around the Falklands' status are, as I said, extremely complicated — if you're interested, Lowell Gustafson's The Sovereignty Dispute Over the Falkland (Malvinas) Islands provides a very good, very detailed primer on the political and legal questions.

The reason war broke out in 1982, specifically, has a lot to do with the domestic situation in Argentina at the time: the ruling military junta, then led by Leopoldo Galtieri, was facing a major economic crisis and severe public unrest.

A very good argument can be made that the invasion of the Malvinas was partly machismo military adventurism by the Argentine Navy, and partly a public-relations gambit designed to distract from a febrile domestic situation by settling an old score; unifying the country by restoring Argentine territory (and thus national pride). In the event, losing the war played a major role in hastening the fall of the junta the following year.

The Malvinas issue remains a populist political touchstone in Argentina — the current president, Cristina Fernández de Kirchner has been particularly vocal about it.

Lawrence Freedman's The Official History of the Falklands Campaign is a very detailed but very readable two-volume history of the war — the first volume deals with the causes and context for the war; the second with the war itself and its outcome. I'd also look at The Land That Lost Its Heroes by Jimmy Burns for an account of the war and its impact on Argentina.

as a follow up question, can I ask; Why did not the US: Monroe Doctrine prevent this from happening? I have asked other people before so I know there are differing opinions.

obviously at the time, US president Reagan had a close relationship to Thatcher. But my question is more on theoretical grounds.