Ant - a Germanic term, seen in Old English aemette which is why it is still known in the southwest of Britain as the 'emmet', and coming from the Germanic root ai 'off, away' and Proto-Indo-European mai 'to cut' (a word which has also given us 'maim'). Thus the ant or emmet is actually 'the biter-off', a reference to how the creatures chop up larger prey to take back to their nests.
Aphid - the gardener's traditional enemy has an odd name which has never been understood. It is unknown before 1758, the colloquial name of ant-cow from 1847, when it was apparently coined by Linnaeus as aphides, the plural of aphis. Here the trail goes cold, although it has been suggested this comes from Greek apheides meaning 'unsparing, lavishly bestowed' and a reference to its unbelievable rate of production. While this etymology seems plausible there is nothing to show it to be true. What is true is their prodigous breeding capability for, under optimal conditions and with no predation, disease. parasites, and unlimited food supplies, a single female aphid can, through asexual reproduction, theoretically produce 600,000,000,000 (six hundred billion) descendants in a single season. Now you see why insects have been suggested as the answer to the world's food problem - although it doesn't answer the question the aphids ask about their food problem.
Bee - since humanity's days as hunter-gatherers honey will have been an important natural resource. Freely available, albeit not easily gathered, nutritious, sweet (and who doesn't like sweet), and with a long shelf life, it was the answer to many needs. Hence there has been a long association with the bee and that is reflected in the name. Coming from Old English beo, Old Norse by, Old High German bia, Middle Dutch bie, Proto-Germanic bion, and Proto-Indo-European bhei, all mean simply 'bee' and thus the name of this now worryingly endangered insect has not only never changed but has always been imitative of the buzzing sound associated with it.
Beetle - from Old English bitela, itself from Proto-Germanic bitel 'biting', and ultimately traceable to the Proto-Indo-European bheid 'to split' and referring to the formidable mandibles of beetles.
Caterpillar - the larval stage of butterflies and moths does not have a name it can call its own for, no matter what it is known as and from what language, the creature is always alluded to as resembling something else. For example, the English 'caterpillar' came from Old French chatepelose or 'shaggy cat'. Nothing different in other tongues, Swiss German teufelskatz 'devil's cat', Milanese cagnon 'little dog', Italian gattola 'little cat', Portuguese lagarta 'lizard', and Kentish where it was either a 'hop-dog' or 'hop-cat'.
Centipede - simply unites Latin centum and Proto-Indo-European ped 'foot' - although known centipedes have anything from 30 to 354 legs (or 15 to 177 pairs) and always an odd number of pairs of legs and therefore no centipede can have a hundred legs (see also millipede).
Chafer - a kind of beetle taking its name for a similar reason. Here Proto-Germanic kabraz meant 'gnawer' and came from Proto-Indo-European geph 'jaw, mouth'.
Cricket - comes from the Old French criquet from criquer, which means exactly what it sounds like 'creak, rattle'.
Dragonfly - is a fairly modern name, dating from the early 17th century and an example of folklore more than its earlier name of adderbolt, a good description of its shape and movement.
Earwig - named because it was thought the the wicga 'beetle, worm, insect' would be likely to hide inside the human ear - French perce-oreille and German ohr-wurm give the same warning - although there is not a single recorded instance of any earwig found in any earhole throughout the entire human history. The term wicga shares an origin with 'wiggle' in Proto-Indo-European wegh 'to go, move'. Also worth noting is the old dialect term from the north of England, where it was known as a 'twitch-ballock'.
Flea - see 'fly' below.
Fly - this is from the sense of movement through air, but this comes from the word 'flee' (as is 'flea' above) as it was the fastest means of escape. Here 'flee' comes from Proto-Indo-European pleuk, from the root pleu 'to flow'.
Gnat - shares a root with 'gnaw' as this means 'biter' but understood as 'little biter'.
Grasshopper - is basically the same thing as a locust, except from time to time the locust form vast swarms. The name is obvious, unlike the locust (see below), referring to its movement and habitat.
Greenfly - obviously some are indeed green and they are capable of flight.
Grub - derived from the verb and thus 'the digging insect', in turn this is from Proto-Indo-European ghrebh 'to dig, bury'.
Hornet - there is no doubting this is from 'horn', but whether this refers to the instrument and thus the buzzing sound of the creature or to the sharp feel of its sting is unknown. Interestingly as the first instruments known as horns were likened to the shape of the animal horn, both share an origin.
Katydid - named for the sound made when the male vibrates its wings. Of more interest is the alternative name for the insect, again imitative of the males but said to sound more like "Katy didn't".
Ladybird - or in the USA the ladybug, the latter are far more sensible name as it clearly is not a bird - although better still was the earlier name of 'ladyfly'. The 'lady' here is the Virgin Mary, which can be seen better in the German Marienkafer.
Locust - shares an origin with the name of the lobster, indeed the French form is languste and the Latin locusta meaning both 'locust' and 'lobster'. Now onbviously it referred to any multi-limbed creature with an external skeleton but, other than that, the etymology is a mystery. Note the Latin lacerta is the only other word known to refer to two quite different creatures, in this case the lizard and the mackerel.
Louse - a parasitic insect and one which has been with humankind for so long its name has never changed, at least not since Proto-Indo-European lus.
Mantis - often referred to as a 'praying mantis', where the first element is superfluous as the name 'mantis' comes from the Greek mantis meaning 'one who divines, prophet'.
Millipede - as with centipede (see above) this combines Latin and Proto-Indo-European to give 'a thousand feet', although the most ever discovered had 750 feet.
Mite - can be traced to Proto-Indo-European mei 'small', something which is seen as another meaning of the word 'mite' today. And before anyone points this out, I know a mite is an arachnid, not an insect.
Mosquito - is a Spanish word, itself derived from Latin musca 'fly' and ultimately the Proto-Indo-European mu 'gnat, fly'.
Nit - little change in this, the egg of the louse (see above), since Proto-Indo-European knid which referred to exactly the same thing today.
Spider - an obvious name when we realise this can be traced back to Proto-Indo-European spen meaning 'stretch, draw, spin' and thus the spider is literally 'the spinner'. And before anyone points this out, I know a spider is an arachnid, not an insect.
Tarantula - named from the seaport city of Taranto in southern Italy where these spiders are frequently found, as a place name it is thought to come from darandos 'oak trees'. And before anyone points this out, I know a trarantula is an arachnid, not an insect.
Termite - began as the Latin terere 'to rub, erode', then termes 'woodworm, white ant', and then to Modern Latin termites, pronounced as three syllables: 'ter-mi-tees', which was mistakenly thought to be plural and the final 's' dropped to produce an apparent singular.
Wasp - no matter how far back we trace this word it, like 'bee' (see above), has only ever meant 'wasp'. It is likely related to webh meaning 'to weave' and a reference to the production of the nest.
Weevil - exactly as 'wasp' (see above) in coming from Proto-Indo-European webh 'to weave'.