I found a discarded Mongoose Alta frame which has an front wheel marked Araya - Japan'.
The wheel reads 26 x 1.50 HEGP710, what rear wheel should I get?
Any regular mountain bike wheel with diameter 26 inches, rim width between 20-30 mm and a freewheel would likely to work. Cheap wheels in this category cannot be too wide for you, and absolutely most of them have 135 mm hub width and QR skewer to attach the wheel, which should be what your frame's rear dropouts will accept.
You will also need to buy matching tires (anything from 26" × 1.5" to 26" × 2.1" to your liking), inner tubes (almost any marked as 26" will do) and a cassette that fasts to the wheel's freehub. The choice of cassette depends on how many gears your rear derailleur has; for an 8-speed derailleur you will need a cassette with 8 cogs etc.
I'm guessing that '26 x 1.50 HEGP710' is what is written on the front tire, not the wheel.
The Mongoose was not a high end bike, so you just need an inexpensive replacement rim brake mountain bike wheel.
Rim diameter is 26" or ISO 559. The tire is 1.5" wide so you need a rim width that will handle that: 21-23mm will do. You can also measure the internal distance between the rim flanges on the front wheel if you want an exact match.
The hub is very likely 135mm wide. The axle type needs to be a quick release. Measure the distance between the internal faces of the rear wheel dropouts if you want to be sure.
Update: you will also want to go with a spoke count that matches the front wheel. A quick look at a Google image search for 'Mongoose Alta' shows the bike came with 32 spoke wheels.