test/
run1.log  run2.log  run3.log

##</code><code>##

test2/
run2.log  run2.log_1  run2.log_2

##</code><code>##

use warnings;
use strict;

use File::Basename;
use File::Copy;
use File::Find::Rule;

my $count = 1;
my $dir = 'test';
my $to_dir = 'test2';

my @files = File::Find::Rule->file()
                            ->name('*.log')
                            ->in($dir);
if (@files){
    for (@files){
        if ($_ =~ /run/i){
            my_copy($_);
        }
    }
}

sub my_copy {
    my ($base_file, $file_name) = @_;
  
    # get just the filename

    my $file = basename $base_file;

    # $file_name will be empty on the initial call...
    # let's set it so it's not undef

    $file_name = $file if ! $file_name;
    
    if (-e "$to_dir/$file_name"){

        # strip off the _N suffix if present

        $file_name =~ s/_\d+$//;

        # append a new _N, then increment the count

        $file_name = "${file}_$count";
        $count++;
       
        my_copy($base_file, $file_name);
    }
    else {
        print "copying $base_file to $to_dir/$file_name\n";
        copy $base_file, "$to_dir/$file_name" or die $!;
    }
}

##</code><code>##

test2/
run1.log  run2.log  run2.log_1  run2.log_2  run2.log_3  run3.log

##</code><code>##

test2/
run1.log run1.log_1 run2.log  run2.log_1  run2.log_2  run2.log_3  run2.log_4  run3.log  run3.log_1