doc says it can be used to rename "members", it doesn't actually show how

It's a little hard to find, see the section "Member Operations":

fileName() / Get or set the member's internal filename.

This works for me:

use Archive::Zip qw/:ERROR_CODES :CONSTANTS/;

my $zip = Archive::Zip->new();
$zip->read('/tmp/foo.zip')==AZ_OK or die "read";
for my $member ($zip->members) {
    next if $member->isDirectory;
    my $fn = $member->fileName;
    $fn =~ s/foo/bar/;
    $member->fileName($fn);
}
$zip->overwrite==AZ_OK or die "write";
[download]

Update: My 1111th node! :-)