Although there are several solutions for your task, consider the following:

use warnings;
use strict;

my $names = 'names.txt';
my ( $found, $name );

print 'Enter a name: ';
chomp( my $input = <STDIN> );

open my $fh, '<', $names or die "Cannot open $names: $!\n";

while ( $name = <$fh> ) {
    chomp $name;

    if ( $input eq $name ) {
        $found++;
        last;
    }
}

close $fh;

if ($found) {
    print qq{The name "$input" was found.\n};
}
else {
    print qq{The name "$input" was not found.\n};
}
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Note that:

• The name to search for is requested first.
• chomp is used to remove the trailing input record separator (usually \n), from both the entered name and the names from the file.
• The names' file is read once; no hash or array is necessary to 'hold' the names (Why process the list twice?). A flag's set and the while loop is immediately exited if the name is found.
• Appropriate responses are given depending upon the value of $found.

Hope this helps!