I want to add a caveat to this, if I may. Although @mdewey's answer is widely accepted, the precise etymology is uncertain, although it almost surely comes from Nahuatl. There is little historical evidence of the "many trees" theory (the Wikipedia's article source is a blog on the "rainforest facts"). I don't think there is any historical evidence, to be honest.
An alternative theory, supported by ethnohistorical accounts, is that the name is derived from the Nahuatl word for "eagle"
cuahuitl (as oposed to
cuāuhtli, "tree"). This referred to the "eagle warriors", the elite military unit at the city of Iximche. Iximche was the capital of the Kaqchikel people and effectively served as the first Spanish capital of the country. Below you can see the city represented via its "eagle warrior". (This comes from the Lienzo de Quauhquechollan.) I believe this theory is generally more popular amongst archaeologists and has, all in all, more evidence to support it.
I can also add a thing or two about the word, being a Guatemalan myself. The first is that we consider it (unsurprisingly) a very rude expression. The second is that the same decomposition of the word
Guate-something has been widely used.
Guatebuena is common, and slightly cheesy.
Guate-morfosis was a recent, Pepsi-financed, campaign.