Always singular after "any". So any other member is the correct one. "just make a sound, any sound will do", " any day will be fine", "any person found in violation of the school regulations will be expelled". If you notice, "any OF THE other memberS" is of a different grammar structure. Here the last part is left out and is understood because of "the" to indicate the "group"... She is shyer than any OF the other member[S] [OF THE GROUP]. If you use singular, it pertains to one person, and if plural, comparing to several people in the group.

Hey Tiger, Here you have 2 options. 1. She is shyer than any other member. 2. She is shyer than any of the other members. The first one implies that she is shyer than any of the other single members, by using the singular word, member ,we take all the posible options and categorize them under one word. For the second option we use the plural form of member because we are also using the preposition "of" indicating there are many options from one group, so OF ALL THE MEMBERS she is shyer than any of them, which also implies that is the shiest. Is a little confusing but I am happy to further explain!

She is shyer than any of the other members. Or: she is shyer than any other member (of the group)