Glynn, glad you're interested in making them. They're quick, functional and make good quick gifts you can hand out to family and friends.
Wood - dry and stable. I've made them out of pine stock stacked up and glued together, high grade plywood stacked and glued, solid pieces that are big enough, etc. Whatever you have works. It's a mug so unique and different are good. A good minimum for OD on the top is 3 5/8, or thicker if you like. If you make it really thin you can get it out of a stack of 1x4 scraps glued together but the top wall edge will be thin. I've done it but the top edge walls are really thin and can break or chip. Thick is OK, it's a mug sorta. Height - most of mine are about 4 3/4 tall so plan for a tenon and some waste to shape. A 6" high stock piece is plenty.
Finish - I'm a walnut oil finish guy but anything works really. Some cutting board oil, danish oil, up to you.
Process tips - drill a large hole first with a Forstner bit all the way to the bottom. I use a 2 1/4" bit just to save time and set the bottom inside dia. You can pick one up on Amazon for a few bucks, not expensive, and saves a lot of hogging out time. If you don't have one just drill the biggest hole you can to give you some room to work. The hole is deep so will need large easy wood tools or be careful setting your tool rest inside the cup as you clear out the material or you'll over-extend the smaller tools and fight it. Don't sweat making the inside a perfect fit to the guide or paper cup. Clean and even look nice and as long as the cup doesn't wiggle around and sits snug who cares that it doesn't touch all the sides. Better if it doesn't actually. The critical area to watch is the top lip. The guide should be just a grunt (TN tech term) below the lip. Test a cup from time to time, or a few since they are not a precision item, and you'll get a sense of how snug it feels. I sometimes shape the outside last. Mugs can be tapered, straight, fat, curve out, etc. Up to you. Just make sure you can easily pick it up to use it.
Tools - you're fine and it's what I use to make them. I started with the small set of easy wood tools and still use them. I have the big set and use them as well. Handy to have for certain tasks and you can go a long way with them. You can tilt the round cutter a few degrees counterclockwise to clean up inside wall cuts and use the square one to clean up the inside bottom rim/edge as needed. I leave the pinhole in the bottom from the Forstner bit most often. Not worth dealing with.
Let me know if I can help along the way.
Stacked plywood. It's what I'm using this morning. I have three I keep handy. Plywood can work but it's hit or miss and needs to be a very good grade.
. Would you mind if I impose on you and ask a few questions?