THere are several issues you need to correct.

First - you need to fix how you fech the param value - at the time the form' s values are being collected, $unit_name does not exist - so you need to do:

my $selectedName = param('unit_name');
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Next, you need to create a single hidden field OUTSIDE the loop:

 print "<input type='hidden' name='unit_name' value='This gets set via
+ javascript'>";
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Next, your loop should look something like this:

    print "<td>\n"; # If you want all buttons in a single TD .. otherw
+ise, this could be <tr>
    while (my @data = $sth->fetchrow_array()) 
    {
        $unit_name = $data[0];
        push(@unitNames, $unit_name);
        print "$unit_name<br>";
        print "<input type='image' name='$unit_name' id='$unit_name' a
+lt=airConButton src='../../images/unit.jpg'
          height='100px' width='100px' onclick='this.form.unit_name.va
+lue =\"$unit_name\";this.form.submit()'></input>";
    }
   # Close the TD or TR here..
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