A sledgehammer to crack a nut
To use 'a sledgehammer to crack a nut' means to use disproportionate force or expense to overcome a minor problem.
Sledgehammers are large iron hammers which were first used in England in the 15th century. These weren't tools to hammer sledges - the little ice trolleys with runners that the young Citizen Kane was so fond of. 'Sledge' was the original name of this form of hammer; so 'sledgehammer' is something of a tautology. 'Sledging' has recently reappeared as a verb form in the previously refined and gentle world of cricket, where it means the browbeating and harassment of the batsman by the fielders.
'Sledges' were an English invention but this phrase wasn't - it first saw the light of day in 1850s America. 'A sledgehammer to crack a nut' is one of the many versions of the phrase, the others having faded into disuse. The spelling of 'sledgehammer' hasn't yet settled down and is still making the usual progression of hyphenated words pairs, that is, from 'sledge hammer' to 'sledge-hammer' and eventually 'sledgehammer'. The OED prefers the hyphen, but 'sledgehammer' is now the more common spelling.
Pretty well anything which is small and easy to squash has come verbally under the hammer, typified by nuts and insects. These have included peanuts, walnuts or just nuts; also gnats, flies, mosquitoes etc. The first to fall victim was the humble fly, as in this piece from The Gettysburg Compiler, June 1878:
"Don't worry over little ills of life. It is like taking a sledge hammer to kill a fly."
Nuts came into the picture a little later, specifically peanuts; for example, this from The Reno Weekly Gazette And Stockman, May 1893:
"We know some men who are always looking for a sledge hammer to crack a peanut."
Insects and nuts seem to have become combined in the later 'sledgehammer to kill a gnat' version; for example, Grosvenor B. Clarkson's Industrial America in The World War, 1923:
"The Board never used a sledgehammer to kill a gnat."
Oddly, although the common form is now 'a sledgehammer to crack a nut', the first examples of that in print date from as late as the 1950s. Whether our current usage derived as a rewording of the 'crack a peanut' or the 'crack a gnat' version isn't clear - perhaps something of both?
See other phrases that were coined in the USA.