On a more seriousness note, you are still lost half a year later, so I will offer some guidance.
You seem to have a be missing quite a bit of information on electronics. The questions you are asking are lacking some of the fundamental prerequisite knowledge for building a gamecube portable.
I am not saying this to insult you, or make you not want to post again. I just don't want you to get hurt. Li-ion batteries are safe in controlled situations, but are
incredibly dangerous if proper safety precautions are not met. This is probably <10whr in capacity, but portables can have batteries as big as 50 or 100 whr. The same effect can happen if charged incorrectly or shorted.
Check out some youtube channels to learn this stuff quickly, such as
Afrotechmods, who will teaches you exactly what to do, or
ElectroBoom, who teaches you exactly what not to do.
Ill help you out on this one though.
First you need to gather all your electronics and measure how much current they draw all together, and convert to watts. Then you calculate whr needed based off of how long you want to power said portable. Figure out the whr, figure out what voltage you want the source to be, and then use ohms law to figure out the mah.
I didn't know any of this a year ago, and I really know what its like to be in your shoes. Modretro is unmatched for helping with videogame console specific electronics, but I assure you it is MUCH easier to learn basic electronics on other websites. So go learn the basics the easy way, and then start digging for specifics.
If you need any help, I will gladly try to answer any question you have or point you to where to find the answer. I look forward to seeing your portable!