Torque Hinge, Pivot Joint, or ? 

Hinge Types – A Modern Overview — HardwareSource.com

Types of Hinges - Monroe

You really need to learn how to learn.

Have you done ANY web searches before coming here?

Any reading about mechanical design?

I've been doing web search for a couple of days now. But now I feel more lost than when I started it. Mainly because I don't know what options are available and what their names are. I found one book about designing mechanical joints but it was too general and not so helpful. So, yes, I do need to learn how to learn:/.

Do you know any book or reading material that helps me understand the difference between the function of different joints or hinges available for rotational movement?

You should be able to find something here that will work for your budget...

spring loaded lid support

This design might work with a little tweaking....counter balancing the weight with a spring, and locking in an upright position....this way the top could be lifted with a small handle....and then a latch to close securely...

http://www.hardwaresource.com/hinges/specialty-hinges/chest-hinges-box-hinges-lid-supports/toy-box-lid-supports/bench-seat-hinge-1

https://www.amazon.com/Sonline-Spring-Loaded-Stainless-Toggle/dp/B00X9HBR5S/ref=sr_1_9?ie=UTF8&qid=1479615180&sr=8-9&keywords=Snap+Latches

That lever off to the right looks like a storage nuisance.

Forget rotary dampers - you'd need a huge one to do this little job (consider the torque!). Way over-engineered/over-priced. Use a linear damper.

Look for shoulder screws and work out how you could use them. Look up finger joint and think about using one of them for the centre pivot. As for damping, use nylon6 washers in the finger joint and tighten the screw/bolt until desired damping is achieved. Look up brass bush, stub axle and tapered roller bearings. Have fun