Starting a new game in Elite Dangerous

You can delete your save from the Main Menu, per the FAQ:

To reset your game, go to back to the Main Menu » Options and the Clear Save option will be available. WARNING: Selecting this will delete all in-game progress. If you choose to do this, the Create New Player screen will appear and you will be able to choose a new name.

If you just want to reset your position, you can kill yourself. If you die, you get the "Ship Destroyed" screen. This allows you to pay your rebuy cost (5% of the total value of your ship and loadout) to get your old ship back, and you will spawn at the last station that you docked at.

You will also notice another option, "Free Sidewinder". This is a free ship that is always available, in the event that you cannot pay the rebuy costs for your old ship. This free Sidewinder is your starter ship, and choosing this option will spawn you at the starter location. This restarts your ship and location, but keeps your credits, rank, etc...

In the menu on the right, there should be an option to self-destruct, which will spawn you back at the last station you visited.