I have something akin to and need to change the single quotation marks to double quotation marks. I tried :s/\\\'.*\\\'/\
:s/\\\'.*\\\'/\
You've got the right idea -- you want to have "\1" as your replace clause, but you need to put the "Hello There" part in capture group 1 first (0 is the entire match). Try:
"\1"
:%/'\(.*\)'/"\1"