I have commented some lines in your program and added some others as follows:

#!/usr/bin/env perl

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

my $source = shift ;
my $lines_per_file = shift ;

 open (my $FH, "<$source") or die "Could not open source file. $!";
# open (my $OUT, '>', '00000000.log') or die "Could not open destinati
+on file. $!";

my $OUT = undef;
my $i = 0;
#my $index_last = 0 ;
my $index_current = -1;

while(my $line = <$FH>) {
    next unless ($line =~ /mrule/) ;
    if ($line =~ /mrule=([0-9]+)/){
        if( $index_current != $1 ){
            $index_current = $1;
            if( defined($OUT) ){ close($OUT); }
            my $NEW = sprintf("%08d", $index_current);
            open($OUT, ">${NEW}.log") or die "Could not open destinati
+on file. $! " ;
        }
        print $OUT $line;
        $i++ ;

#        if ($1 != $index_last){
#            $index_current = $1 ;
#            close($OUT);
#            my $NEW = sprintf("%08d", $index_current);
#            open($OUT, ">${NEW}.log") or die "Could not open destinat
+ion file. $! " ;
#            }
#            $index_last = $index_current ;
    }
}

close($FH);
#close($OUT);
if( defined($OUT) ){ close($OUT); }
[download]

The above will be looking for an mrule=[0-9]+ pattern in the input. Once it finds one, it will check if the current index is the same as the one in the line and if not, it will close current filehandle and open another one with the new name. After that, it will print to the filehandle currently opened.

Note that no filehandle is opened unless the pattern in the input appears.

tested with the minimal file you had provided.

bliako