Formerly 曬, which consists of 日 sun and 麗 beautiful. The newer and simpler 晒 (using 西 West) may have been an attempt to restore the phonetic or just an convenient abbreviation. Its original meaning seems to have been to dry in the sun or to expose to the sun, with later on the figurative meaning of to expose as an extended meaning.
Mnemonic: Exposure to the Western sun has a bleaching effect.
晒 is a modern variant for 曬. The Shuowen tells us that 麗 is its phonetic. However, it doesn’t seem to be a predictable phonetic at all for more recent Chinese, which might be one reason for substituting the phonetic for the easier 西 (which is an only slightly better phonetic to be honest).
Actually the meaning of 麗 could make more sense than it pronunciation: make something beautiful using the sun (leading to bleach) - but that’s just guessing.
When I first looked up 晒す in a Japanese dictionary (sarasu, ‘to expose; to bleach’) I mistook 西 for 酉 thinking that a verb with the meaning to bleach would be related to 酒 alcohol-like substances to treat stuff to bleach them. 酒 could be a twisted mnemonic in that way, but it’s wrong unfortunately, 西 is simply ‘the West’ - and I don’t expect a Western sun to have any more bleaching or exposing effect than an Eastern sun. Surely it’s just a convenient abbreviation and phonetic.
Still, as a mnemonic one could come up with could be something like: exposure to the Western sun has a bleaching effect.