nearly all wood arcade cabinets are 3/4" particle board. standard t moulding is 3/4". no reason to use anything else. the only thing i've seen NOT 3/4" is the tops of cocktail tables which are usually 1 inch.
as for materials some people prefer 3/4 MDF while others go the plywood route.
if you want to eschew the t-moulding, and route the edges again, MDF is more consistent all through the entirety of the material and would leave a nice edge.
if you want to just whack something big together, a good one side plywood is plenty strong, and doesn't need much prep to paint or anything like that, and could even be stained.
when it comes down to it, personal preference and ability to work with a material are basically the limiting factors.
I personally buy sheets of 3/4 black laminate particle board that are laminated on both sides. you have to be careful not to scratch or gouge it and need to use a sharp blade when you cut it so the edges don't get chipped... but the result is a nice surface that needs nearly zero work in finishing.
I think if I where to use something less than 3/4" would be a mini or micro sized build... and i'd probably go 1/2" with it. not that there is anything wrong with using 3/4" but more to keep the weight down and maybe give a tiny extra bit of space inside if I were trying to keep a certain dimension outside.