I've used tilemill to style my shapefiles and export it as mbtiles file. The intention is to run the map application offline. What's the best way to serve the tiles from the .mbtiles file to openlayers? I see al lot of tileservers on the internet but i can't figure out what way is the best lightweight solution is for me.
There is not best lightweight solution. It's more about finding a solution that fit with your constraints. Here some questions to help you make a choice.
Do your company or yourself only use a particular language? If yes, search among the solutions for your language. If no, just take one you will be able to deploy at least.
What is your hosting? Do you have a dedicated server or not? If not, just use supported languages from your hosting (LAMP, I'm looking at you)
Do you only plan to use tiles images or do you plan also some sort of interactions? All MBTiles servers can serve images but not all of them can serve UTFGrid JSON files, responsible for interactions.
Last question is about maintenance/freshness. Now, the spec evolves but not so much, so you may choose depending on the date of last update to the code (if there is code, you know more less with DVCS when it was last updated)