If you are "just" doing quick snapshots of your family, the dogs, the tractor, a project, etc, then any decent point and shoot will work. The thing about photography equipment is the more you pay the more you get. It really is a question of will you use what you get to make what you pay worth it.
I assume you have a 50mm F1.8 lens on the old FE camera. That lens will very likely work on many of the modern Nikon DSLRs. You won't get auto focus but I would guess a good many of the new cameras would support the lens.
I have five lenses four of which I bought in the 90s for film cameras. I bought a 50mm lens a few years ago. One thing to be careful of in DSLR's today is that the full sensor which is the size of 35mm film is only in the high end cameras. Most cameras are cropped sensors which change the perspective on the lens from a full frame sensor. A 50mm lens becomes 75 on a smaller sensor camera. The reason I brought this up is that there are lenses being built that are for the smaller sensor. Moving up to a full frame sensor on those lenses will be problematic. But the older film lenses will work just fine on the cropped or full frame cameras.
The other reason I brought this up is that DSLRs are NOTHING more than computers to attach lenses. In 18-24 months there will be another DSLR out for the same money with even better features to temp you out of your money. The lenses, not so much. Nikon has "improved" one or two of my lenses since the 90s. Nothing they have changed has made me spend more money on the latest and greatest lens. Lenses hold their value while camera's loose value quickly.
Unless someone is taking photos in extreme conditions like low light requiring fast shutter speeds a high end DSLR is not needed. Spend more on the glass not the camera. Glass Lasts. Right now I am taking photos of kids basketball games. The gym is very well lit for a gym but it still is taking a very good camera and the professional lenses I have to get the photos. However most of my photos are being taken with a 50mm/F1.4 and 105mm/F2.8 prime lens with the aperture set to F2.8. The 50mm/F1.8 lens would do just as well set to F2.8. It is $124 which is dirt cheap for good glass.
Good lens do help make good photos. I have shown some of the photos to people at work and they have noticed the "pop" in my photos that they don't get in theirs. This is due to the camera, the photo paper, knowing photography fundamentals, but I think the big reason is the lens.
So you could buy a low end DSLR with a cheap nifty fifty. The good thing about digital photography is that you can do all of your own processing at home. This is also a bad thing. Do you WANT to process RAW/NEF files into JPGs? Do you want to learn a photo editing program to get the best image possible? Do you just want to take the danged picture and be done with it?
Later,
Dan