<?php
$a=0;
while($row = mysql_fetch_assoc($execute)){
    $ID = $row['id'];
    $name[] = $row['names'];
    echo "<button name="btn$ID" class='btnName' onClick='showForm()'>$name[$a]</button>";
    $a++;
}

<div id="result" style="display: none;">
    //name from clicked buttons should be displayed here
</div>
?>

<script type="application/javascript">
    function showForm(){
        document.getElementById('result').style.display="block";
    }
</script>

function showForm(ev) {
  const name = ev.target.getAttribute("name");
  const result = document.getElementById('result')
  result.innerHTML  = name;
  result.style.display = 'block';
}