None of the solutions here worked for me because I was on a system that didn't have access to the perl rename script, plus some of the files included spaces. However, I found a variant that works:
find . -depth -exec sh -c '
t=${0%/*}/$(printf %s "${0##*/}" | tr "[:upper:]" "[:lower:]");
[ "$t" = "$0" ] || mv -i "$0" "$t"
' {} \;
Credit goes to "Gilles 'SO- stop being evil'", see this answer on the similar question "change entire directory tree to lower-case names" on the Unix & Linux StackExchange.