You may need to provide more of a context to figure out what is happening.  And I'd also suggest you consider adding use strict; and use warnings to help you in your debugging task.

I was unable to make it fail with the following slight rewrite of your program:

use strict;
use warnings;

use IO::Socket::INET;
use threads;

my $port = 19999;

my $ServerSocket = new IO::Socket::INET (
    LocalPort => $port,
    Listen => 2,
    Type => SOCK_STREAM,
    Reuse => 1,
    Proto => 'tcp'
);

while (my $clientSocket = $ServerSocket->accept()) {
   my $thr = threads->create(\&child_thread, $clientSocket);
}

die "END ServerAccept(), $!";

sub child_thread {
    my ($sock) = @_;
    print "Accepted client connect\n";
    print $sock "Hello client!\n";
    chomp(my $from_client = <$sock>);
    print "Client says '$from_client'\n";
    for (my $i = 0; $i < 5; $i++) {
        # Sleep for a while
        print "(Still in child thread)\n";
        sleep 1;
    }
    print "(Child thread finishes)\n";
}
[download]

And here is the client code client.pl I used to test it:

# Strict
use strict;
use warnings;

# Libraries
use IO::Select;
use IO::Socket;
use Sys::Hostname;

# Outgoing port
my $port = 19999;

my $host = hostname();
my %params = (
    'PeerAddr' => $host,
    'PeerPort' => $port,
    'Proto'    => 'tcp',
);
my $sock = new IO::Socket::INET(%params);
$sock or die "Could not create socket ($@)\n";

# Declare succcess connecting
print "<Connected to server '$host' at port $port>\n";

# Read response from server
chomp(my $text = <$sock>);
print "Client gets text '$text'\n";

# Send text to the server
print $sock "Hi Server!\n";

# Wait for a few seconds
sleep 3;

# Close the socket
close($sock);
[download]

When I run your program, I'm then able to connect using client multiple times without getting any errors:

C:\Documents and Settings\liverpole> server
--- Server ---
Accepted client connect
Client says 'Hi Server!
(Still in child thread)
(Still in child thread)
(Still in child thread)
(Still in child thread)
(Still in child thread)
(Child thread finishes)
Accepted client connect
Client says 'Hi Server!
(Still in child thread)
(Still in child thread)
(Still in child thread)
(Still in child thread)
(Still in child thread)
Accepted client connect
Client says 'Hi Server!
(Still in child thread)
(Child thread finishes)
(Still in child thread)
(Still in child thread)
(Still in child thread)
(Still in child thread)
(Child thread finishes)

--- Client ---
C:\Documents and Settings\liverpole> client
<Connected to server 'golux' at port 19999>
Client gets text 'Hello client!'

C:\Documents and Settings\liverpole> client
<Connected to server 'golux' at port 19999>
Client gets text 'Hello client!'

C:\Documents and Settings\liverpole> client
<Connected to server 'golux' at port 19999>
Client gets text 'Hello client!'
[download]

What happens when you run the above programs?  If they work, see what happens running my client program with your server and see what happens.

If all else fails, please provide more of the context of your own program.