If you're using ColdFusion attribute required, I'm not sure how to change that dynamically. But if you're doing client side validtion with javascript - or double checking it after submitting with ColdFusion, this is easy in principle.
<script language="Javascript">
function checkifother()
{
myselectfield = document.getElementById('myselect');
explanationdiv = document.getElementById('explanationdiv');
if ( myselectfield.value == 'Other')
{explanationdiv.style.display='block'}
else
{explanationdiv.style.display='none'}
}
</script>
<cfform name="myform">
<cfselect name="myselect" id="myselect" onchange="Javascript:checkifother();">
<option value="Option 1">Option 1</option>
<option value="Option 2">Option 2</option>
<option value="Other">Other</option>
<option value="Option 4">Option 4</option>
</cfselect>
<div id="explanationdiv" style="display:none">
<cfinput type="text" name="otherexplanation">
</div>
</cfform>