Best way is to store a hash of both the filehandle and its name:

use strict;
use warnings;
use Data::Dumper;

sub openRandom {
    my $fh = {
        name => (int rand 1000) . '.txt'
    };
    return $fh if open($fh->{'handle'}, ">$fh->{'name'}");
}

print Dumper(openRandom());
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