How to make 5V 650 mA on a summer camp?
First of all, how long must the computer be on?  You mention that it must be activated every 108 minutes, and I assume this means it must be turned on every 108 minutes.  Then for how long does it get turned on for?  At a rate of 650 mA, if it is only on for say 1/2 hour every 108 minutes the you would only use 325 mA every 108 minutes. 

Assuming you follow the 108 minutes 24 hours a day, then you would turn on the tablet 13 times for 1/2 hour in a 24 hour period.  You are camping for 6 days so a total of 13 x 6 = 78 - 1/2 hour sessions (or however long you are turning it on for).

At 650 mA, 78 - 1/2 hour sessions (650 mA x 78 x 1/2) = 25.35 amps

Your Nimh are at 1.2 volts, so put 4 in series to get 4.8 volts (closer to 5.6 volts fully charged so be careful).  When in series, the capacity is NOT added, so have only 2300 mAh capacity with the 4 in series.

25.35/2.300 = 11.02

So you would need 11 sets of batteries, or recharge them 11 times if you use the tablet for 1/2 hour every 108 minutes 24 hours a day for 6 days straight.  I have a feeling that you won't be using the tablet this much, so you will have to adjust your calculations accordingly.

I know that you mentioned 18650 batteries, but given you budget and your options, I think that a good quality Nimh is your best choice. The main reason is because you can get a reasonably priced Nimh AA battery that can be charged in less than 2 hours!  Many lithium ion batteries will take longer to safely charge.  Plus the voltage of 4 Nimh batteries matches what you need.  Get a fast charger that will charge 4 at once.  You can get chargers that will charge Nimh in as little as 15 minutes, but that is hard on the batteries.  I would go for a 1 hour charger with the generator.

Good luck!