And is the moniker an attempt by one side or the other to down play the seriousness of the conflict? 'The Troubles' just seems a very quaint way to describe armed conflict where there was bombs and assassinations going on throughout the conflict.
So this answer isn't actually all that long,
The term troubles wasn't new for Northern Ireland, or Ireland as a whole, it was originally used for the time period of the Irish war of independence and the the strife surrounding it. The term was even loosely used to refer to other times of strife in Irish history, 'times of trouble', 'troubled years' etc. The usage of the term is believed to have originated from Irish funeral etiquette in which people would offer condolences by apologising for the relatives 'troubles'. Funerals in Ireland were and are fairly communal so this would have been decently well spread and well heard.
When news sources began reporting of the violence of 1969 it was frequently compared to the violence of 1921, with one paper carrying the title saying 'The worst flare up since the 1921 troubles'. In reality '69 was fairly bloodless in comparison to '21, the July-August violence of '69 had seen 10 deaths, 8 during the 12th - 16th violence during August, the same period in '21 saw 48 killed, 21 of which were killed in a three day period at the end of August of that year. The old Belfast troubles lasted two years and saw almost 500 people killed in that time frame, the new troubles seemed understated in comparison in the two years (July 69-August 71) since they began 'only' 76 people had been killed, this came to a brutal escalation when, three days from the second anniversary of the battle of the Bogside the British army launched Operation Demetrius, an internment campaign in the start entirely focused on Republicans. 34 people had died in conflicted related instances in the eight months of '71 before Operation Demetrius, four months later that total would be 171. 1972 saw 480 conflict deaths. The comparison was now bloodily apt.
A few years later a Thames television series into the conflict was broadcast on national tv, the show looked at life during the conflict and carried the title 'The Troubles', the name had stuck. Of course the name also had political advantages, 'war' was a dirty but legitimate word, it became the favoured term of both the British and Irish governments to refer to the fight without legitimizing the IRA's violence. It also has the benefit of being neutral, it blames no side for starting the violence as opposed to the 1920-22 conflict which is at times referred collectively as the 'Belfast Pogrom'