Old English byrgan "to raise a mound, hide, bury, inter," akin to beorgan "to shelter," from Proto-Germanic *burzjan- "protection, shelter" (cf. Old Saxon bergan, Dutch bergen, Old Norse bjarga, Swedish berga, Old High German bergan "protect, shelter, conceal," German bergen, Gothic bairgan "to save, preserve"), from PIE root *bhergh- "protect, preserve" (cf. Old Church Slavonic brego "I preserve, guard"). Related: Buried; burying. Burying-ground "cemetery" attested from 1711.
The Old English -y- was a short "oo" sound, like modern French -u-. Under normal circumstances it transformed into Modern English -i- (e.g. bridge, kiss, listen, sister), but in bury and a few other words (e.g. merry, knell) it retained a Kentish change to "e" that took place in the late Old English period. In the West Midlands, meanwhile, the Old English -y- sound persisted, slightly modified over time, giving the standard modern pronunciation of blush, much, church.
c.1300 "small ax" (mid-12c. in surnames), from Old French hachete, diminutive of hache "ax, battle-axe, pickaxe," possibly from Frankish *happja or some other Germanic source, from Proto-Germanic *hæbijo (cf. Old High German happa "sickle, scythe"), from PIE root *kop- "to beat, strike" (cf. Greek kopis "knife;" Lithuanian kaplys "hatchet," kapoti "cut small;" Old Church Slavonic skopiti "castrate").
In Middle English, hatch itself was used in a sense "battle-axe." In 14c., hang up (one's) hatchet meant "stop what one is doing." Phrase bury the hatchet (1794) is from a supposed Native American peacemaking custom. Hatchet-man was originally California slang for "hired Chinese assassin" (1880), later extended figuratively to journalists who attacked the reputation of a public figure (1944).