Filtering list in CP index page for new ElementType
In case anyone else comes up against this, here's what I did to fix it.

In my PluginName_MyElementType class which inherits from BaseElementType, you need to override the defineSearchableAttributes method, returning an array containing all the names of the properties of your Model which need to be searchable. So in my case it's just name. The reason you don't need to worry about adding title is it's already done for you under the hood, it's just that I wasn't using it in this case. 

public function defineSearchableAttributes()
{
     return array('name');
}