Which bit of wood is the best? or 'best fit'. I say best fit because in a project like this you often use what you can get your hands on easy and which will be the best to work with. Straight out of the block I can say 90% of people use MDF for their at home arcade machines. why? Because it's a good price point and you can get it pretty much at any good hardware stores. Lets go over some of the other common types.
The price might look attractive but don't bother. You'll end up with more head aches than your savings.
MDF is a great medium for an Arcade Machine at home, Just as long as you treat it as the material it is. MDF has a habit of swelling when wet/moist. So you'll need to apply primer/sealer to the MDF, One of the most important areas to apply the sealer is on the ends. The reason is that this is a great spot for the moisture to soak in and the easiest, You'll notice this when applying the sealer.
Another reason to apply paint to RAW mdf is for art work, decals don't stick to RAW mdf very well and you'll find yourself with your art work not sticking or falling off.
MDF - Melamine coated
Same as above apart from the artwork problem. The melamine surface is great for sticking artwork to and it's easily found in white and black. two of the great colours for a home arcade.
If you have a extra cash you can try using PLY there's difference types you can try and has a great look.
There's no rule that you MUST USE THIS THICKNESS, but recommendations are that you'd use about 18mm for the sides and 12mm for the remaining parts of the machine. like the control panel, front boards etc.. Saying that with a bar top (smaller build) you can get away with using all 12 to 16mm. Although we have noticed that having thin side panels does look a little odd, it's best kept to thicker boards. Remember to make sure you can also get the T-Moulding to suit. There is differences between metric and imperial.