A distinctly British Cockney phrase, "bugger this for a game of soldiers" is the idiomatic equivalent of "screw it", "fuck it", or "stuff it." ("Bugger", of course, being a mild naughty word in the British lexicon, along with "sod" and "bloody.") It is used mostly as a last cry of capitulation and exasperation before leaving behind whatever activity caused the uproar. kalen informs me it can also be used as a preventative cutoff: "I could go and pay $1000 an hour for Butch Harmon to coach me, spend a million hours on putting greens, and then go win the Augusta Masters, but bugger that for a game of soldiers!"
The phrase finds its origin in the idea that some things are simply not worth the trouble. While children playing "a game of soldiers" might seem fun and carefree, if they were being forced to play soldiers for real they might think otherwise. Thus there is a subtle undercurrent that the situation has escalated from a harmless one to one more fraught with peril. It may also have ramifications about the pointlessness of things when dealing with legitimately necessary work - the menial task of cleaning up a children's room, for example, knowing full well the room will just be messed up again.
The similar phrase "blow this for a lark" ("a lark" being slang for something as frivolous and irrelevant as a "game of soldiers") is perhaps most famous for its use in Terry Pratchett's Discworld series of fantasy novels, along with his other short stories in various forms. Other variations include "sod this for a game of soldiers", "fop this for a game of soldiers", "I don't fancy this for a game of soldiers," "fuck this for a game of soldiers", and "blow this for a game of soldiers."
Source: http://phrases.shu.ac.uk/bulletin_board/18/messages/319.html. Thanks to BlueDragon, kalen, ascorbic, and Demeter for giving me some context, and Zerotime and avalyn for knowing their Pratchett. And now the lot of you sod off, I'm tired of updating this node!