I remember a support topic i read with regard to a problem when you use the name name for one of your post fields, try prefixing your input names.
<input type="text" name="my-name" ...
And see if that helps..
Follow-up #1
I performed a simple test inside a widget..
if( isset( $_POST ) ) {
print'<pre>';print_r( $_POST );print'</pre>';
}
echo '<form action="'.home_url().'" method="post"><input type="text" name="thename" /><input type="submit" name="submit" /></form>';
I see the post variables that have been set by the form just fine, post up more of your applicable code please.
Follow-up #2
If you try the test code i posted above does it work? If it does, then i’d guess at an issue with either form action of the submit input name, try hard-coding the action or changing the submit input to match my test one, and see if either change fixes the issue.
If neither help, next i’d suggest disabling plugins to rule out interferance(a plugin could be unsetting $_POST variables at some point(just a wild guess, but a possibility nonetheless).