This is a little better (but not tested):

NAME:
foreach my $name ( @name_array ) {
  FILE:
    foreach my $file ( @files ) {
        open my $fh, '<', $file
            or die "Can't read '$file': $!";
        while ( my $line = <$fh> ) {
            chomp $line;
            if ( lc $line eq lc $name ) {
                print OUTPUTFILE "$line\n";
                next FILE;
            }
        }
    }
}
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Note a few things:

• Once a name is found in a file, I break out of the loop and go to the next file using next FILE (made possible by the label on that loop).
• I check to see whether open failed, and I die if so.
• I use chomp to take the line ending off the lines. That leaves just the name with no newline.
• I compare names using eq and lc. That makes it case-insensitive, but it won't match partial names to full names (i.e., "Fred" won't match "Frederick").
• I do not read the whole file at once. I read it a line at a time and conserve memory.

The way you're trying to do it with regular expressions means that you'll end up matching partial names with full names (i.e., "Fred" will match "Frederick" and "Liam" will match "William"). If that's what you want, fine. You can say $line =~ /\Q$wanted_name/ and get that (see also quotemeta).

If you want each name to be found only once regardless of how many files it's in, you can change my "next FILE" to "next NAME" to skip over other files once the name is found.

Update: I also Use strict and warnings