Has the elements 㳄, which is a CO character for saliva (with the connotation of desiring something) and 羊 sheep, which may be an abbreviation for the CO character 羑 but conveys either way something good, something desirable (compare also 美). Thus covet something desirable i.e. be envious. You may want to take 㳄 as water 氵and lack 欠.
Mnemonic: Drooling with a watery wide open mouth for the nice sheep that you lack: envious.
##Verbose explanation and references
羨 appears to be a compound ideograph of 㳄 and seemingly 羊. Traditionally it's interpreted as drooling metaphorically and coveting nice things that one sees or is aware of - with 羊 hitsuji ‘sheep’ representing the nice things (compare 美しい). Confusingly, 説文解字 claims that 羊 is actually an abbreviation of 羑, a Chinese character that represents a word or concept for ‘a guide to goodness’ (at least according to the Unihan definition, 説文解字 simply says 進善也; by the way, judging by a quick google it seems that in modern Chinese 羑 is only used in names).
Considering that 羊 is the signific in 羑 (and 久 merely the phonetic) 羊 might just be very predictable for conveying something ‘good’, which makes the detail about it being an abbreviation in 羨 somewhat trivial. Sheep are good! (At least in ancient China.)
The meaning ‘drooling’ is clearly conveyed by 㳄, which stands for ‘saliva’ and depicts water and an open mouth. (In both Chinese and Japanese it seems to have been replaced by 涎.)
欠, the right side element of 㳄, is very frequent in Japanese. Henshall claims it's originally a pictograph of a person with a huge open mouth attached to it. Oracle bone script (the first example on the right side of this page) seems in line with this claim, but the shape of the seal character seems to depict not so much a mouth as vapors emanating from a mouth (compare 气 / 氣 / 気) indicating a persons breath. Maybe that in the course of centuries the scribes reinterpreted 欠. Even so Henshall seems to have a stronger interpretation than the traditional one of ‘breath’ (see also Wieger 99 A). Henshall traces the big gaping mouth as ⇒ wide open ⇒ vacant ⇒ ‘lacking’. Interestingly the word behind 欠 can also be written as 缺, which has a similar etymology. On the left side the container or bottle 缶, and right: 夬 which conveys ‘pull apart; open’ (being a pictograph of a hand letting go a bowstring), arriving at ‘empty; lacking’ as well.¹
羨 is similar to 盗 in which 次 is actually also 㳄. In other words drooling above a 皿 dish of food ⇒ strong desire for food ⇒ strong desire for things ⇒ let's steal those things we desire! Note that in PCR China 羨 is simplified just as 盗, with 次 instead of 㳄: 羡.