The only things you will need in the cockpit are your licence, passport, wallet, spectacles (regular, spare pair, and sunglasses) and mobile phone. Depending on your taste you will also need your stash of chewing gum, spare pens, etc. Many people use some sort of variation on a handbag, often either a very small soft laptop bag, a very small soft briefcase, or quite commonly, a small bag originally intended to hold a camera or similar piece of equipment.
You need this small handbag because your overnight bag will go in the hold, and even in types where that is accessible in flight, your passengers will be astonished if you suddenly get up and go down the back to fetch something from your luggage.
Your overnight bag meanwhile should be as small as you can get away with for the length of trip, because although your passengers will be whisked away in a limo, you will often be carrying your own bag across the airport yourself. It should be soft sided (to make it easier to load in the baggage bay). Ideally it should be small enough to qualify as a carry-on item on an airline, because you can end up logging as many hours positioning on an airliner as you do flying the jet yourself.