You can make your scale better (to a file with a large number of names) by employing some sort of DB. In this case it also makes the code shorter. Here's your program using a tied hash to a SDBM file. All the modules you should need to do this are included with the perl install.

use Fcntl;
use SDBM_File;

my %db_names;
tie (%db_names, 'SDBM_File', './names.db', O_RDWR | O_CREAT, 0666);
print "Enter your name--> ";
chomp (my $name = <STDIN>);
if(defined $db_names{$name}){
  print "Your name was already found\n";
}else{
  print "Name not found. Would you like to add it? (y/n)--> ";
  chomp (my $yn = <STDIN>);
  if ($yn eq 'y') {
    $db_names{$name}=1;
  }
}
untie %db_names;
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