I hate spending tons of money on equipment for photographing the Northern Lights / Aurora but it is also frustrating when shooting the lights with sub-par equipment. To take the best shots you really need a full frame DSLR camera but even for a great used one with a used lens, you are looking at over $1000 so here is the best sub $1000 option on the market right now.
This camera has a cropped sensor, that is, it is the second best sensor size to have, only falling behind the full frame sensors. This camera has fully manual mode, a must for Aurora shooting, plenty of mega pixels with 24.2, and ISO levels from 100-25,600. It also has the new EXPEED 4 image processor which helps deduce noise and allow for cleaner high ISO shots (important for shooting the Northern Lights).
One of the huge advantages of getting this camera is that lens can be bought that are also compatible with Nikon full frame cameras so when you decide to upgrade you can take your lens with you. Speaking of which, you can buy this camera with the kit lens, and that lens should serve you well for day to day use for about $500 here but check the other sellers because as of writing this you can get the body and kit lens refurbished for under $400, with a variety of used Nikon D3400s for around that price point.
Now here is where things get tricky, you need to get a lens that is compatible with the D3400 but if you want an easy upgrade path, you need one that is also compatible with Nikon's full frames. I would suggest starting with something like the Rokinon 14mm F/2.8 lens. It does several things right, first, it is a wide angle lends at 14mm so you can capture a large portion of the night sky. Next, its aperture gets as large as F/2.8, which is large enough to let in enough light for good Aurora and astrophotography shots. You should be able to pick one up for under $400 and a used/ open box one for about $300.
Don't forget extra batteries, a nice case to protect your investment, and a sturdy tripod, preferable with something that is not metal to hold onto because you can frost bite your hands quickly if you grab on to -40F metal!