Edit
Official Mapbox Android SDK is released.
Also, see Mapbox's answer.
To people who downvote, I answered the question before the sdk exists.
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I think Nutiteq looks pretty promising. Their API has a lot of features. Also, their demo projects are inline commented. You can check out this project. Especially, MBTilesMapActivity might be something that you're looking for. The class shows how to work with mbtiles:
// 1. Get the MapView from the Layout xml
mapView = (MapView) findViewById(R.id.mapView);
// 2. create and set MapView components
Components components = new Components();
mapView.setComponents(components);
// 3. Define map layer for basemap
MBTilesMapLayer dbLayer = new MBTilesMapLayer(new EPSG3857(), 0, 19, file.hashCode(), file, this);
mapView.getLayers().setBaseLayer(dbLayer);
...
// 4. Start the map - mandatory
mapView.startMapping();
// 5. zoom buttons using Android widgets - optional
// get the zoomcontrols that was defined in main.xml
ZoomControls zoomControls = (ZoomControls) findViewById(R.id.zoomcontrols);
You can also take a look at my toy project which is modified from their demo. Basically, my app allows the user to input a url of a mbtiles file. Then, it downloads the file and loads it to the MapView.
If you really have to stick with OSMDroid, this might be helpful. I haven't had a chance to try it though.
Hope this helps.