Details about the tile that that was clicked are not available. If you use the $.runRule() syntax, there is some general context information available about the dashboard and who clicked the tile.
But my point was that you can still make the rule dynamic by passing unique parameter values from each tile/hyperlink. In each tile, you would configure either the $.get() or $runRule() call with a parameter with a value to identify the Thing however you see fit.
Example Hyperlink
$.runRule("RULE-ID", { "thing": "MyCoolThing" })
Example Rule
(You could also use an HTTP Trigger if you are more comfortable with that)
The REST API and $.runRule syntax thread linked above is a good resource as it has examples of using parameters. That article also links to another article on how you can access parameters using context variables in a rule.