You cannot use use in this context because it is wrapped in an implicit BEGIN block (see BEGIN, UNITCHECK, CHECK, INIT and END), and thus gets evaluated at compile time and before you could initialize the value of your variable. The usual work around here is to use a string eval, a la:

my $module_name = 'Foo';
eval "use $module_name;1;" or die "Couldn't use $module_name: $@";
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But it complains that I'm calling "new" on an unblessed reference when I do my $object = $module->new(); on the next line.

This should not be happening. The following code works for me (assuming an OO-module Foo in your path):

#!/usr/bin/perl
use strict;
use warnings;
use Data::Dumper;

my $module = 'Foo';
my $module_path = "$module.pm";
$module_path =~ s|::|/|g;
require $module_path;

print Dumper $module->new;
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Note the s|::|/|g to allow non-root modules. Also note that by requiring you are bypassing the import mechanism.