If the goal is to somehow wire in your lightning component to use the offline storage in the Salesforce1 Mobile App, then currently there is no ready-made option.
But in the same fashion that you can build a Visualforce page as part of a Hybrid Remote mobile app, you could do the same with Lightning Components. In this case you would be building your own mobile app, that would request a Lightning Components app, which then used cordova to access device features.
If I was tackling this, I think the first approach I would take would be to fire up a hybrid remote visualforce app using Mobile SDK, then reverse-engineer the visualforce scaffolding and libraries to see if I could get the mobile app to request my lightning app.
If you get that basic scaffolding in place you could then use the SmartStore feature for your offline data.
I do know of some developers who have spoken of this, and as I said, this is not exactly what you're asking for, but it seemed worth mentioning it as an option.