Adding Lightning Component to Salesforce1 Record Page

Not yet. As of Winter '16, the Salesforce1 app uses the standard page layout (based on the viewing user's profile and the record's record type) as defined in the Page Layout Editor, not a Lightning Page that can be edited in Lightning App Builder. The Page Layout Editor doesn't support adding Lightning components.

You can create custom Lightning pages and add them to the Salesforce1 navigation, but they aren't the same as the Lightning Experience pages that you see in the desktop version.

More information on adding Lightning pages to Salesforce1.

Whilst I've not found a way to directly add Lightning Components to a record page, you can use the ability to include Lightning Components within a Visualforce page.

Using Lightning Components in a Visualforce page

You just need to ensure that the VF page is set to be allowed on Salesforce 1 and then you can include it on the page layout (which should appear both on the desktop and on Salesforce 1) or as a Mobile Card (which will only show up in Salesforce 1).