Sure, you can use the Attributes collection:
ListItem li = new ListItem("myText", "myID");
li.Attributes.Add("data-x", "xx");
dropdown.Items.Add(li);
This will give you this HTML:
<select name="dropdown" id="dropdown">
<option value="myID" data-x="xx">myText</option>
</select>
I suggest you prefix your custom attributes with "data-": http://html5doctor.com/html5-custom-data-attributes/