Currently, the "template path hint" feature displays the template name and block class name.
Fabrizio Branca's Aoe_TemplateHints module extends the native Magento 1.x template path hint functionality to add much more information to the hints including:
Block name (very useful)
Parent block names
Here is a screenshot from his blog post, visualizing what this could look like:
I propose that this functionality be built into Magento 2. If Magento accepts this proposal, I may be willing to build this functionality and then issue a pull request to have it incorporated into Magento 2.