I'm looking to upgrade my wife's Raleigh Superbe 3-speed to a hub with a wider range/more gears. We live on top of a hill and the range of the Sturmey hub doesn't quite cut it for her.

The only problem is that the bike has 114mm rear dropouts. It's steel so obviously it can be spread, but going to 130mm would be pretty extreme. Are any of the 7- or 8-speed hubs available in a narrower spacing?

Sturmey Archer 8 Speed hub XRF8 is available with "Over Locknut Dimension = 116.0mm". (The hub is best for small wheels though, first gear is direct drive.)

I had/have the same issue with my old Tote/Cycle. From my investigations, the Sturmey-Archer 8 was the only way to go. I believe there was also a SA 5-speed that would fit, but the price between the 5 and 8 was small enough that I didn't really consider the 5 an option.
But, in my case, I ended up staying the 3 speeds because I needed the coaster brake and while I think there was a SA 8 with a coaster brake, the addition of the brake made it too wide.
So-- no integrated brake: I think SA8 is an option. Integrated brake: maybe an SA5, but I don't know for certain. The price jump between 3 and 5 was such that I didn't really think it worth investigating.

I don't need an integrated brake, so the SA8 would fit. But with 1st gear being direct drive, it's going to be hard to get a reasonable hill-climbing gear.

There's an SRAM 5-speed hub that's 122mm, which is close enough. But it would be nice to have more than 5 speeds.

Nah,I was tossing around the idea of putting one of those on my Otis once,and did some tinkering on Sheldon's site for gearing. You can go small on your front ring and they should go up to 21T for rear sprockets for that hub. You could always go with MTB cranks and go down to some pretty small front rings.

On the Sturmey 8, with a 25t rear sprocket, and a small front chainring, you'll be OK.

Really, I'd suggest the Sturmey 5 instead. It is more durable, more efficient, lighter, it uses standard sprockets, and the gear range is only slightly smaller than the 8.

The Sram 5 costs more and has the annoying clickbox, so I'd go with Sturmey in the 5 speed.

The bike in question is a '73 Raleigh Superbe, complete with cottered cranks and 26 TPI bottom bracket threads. So changing cranks is an adventure in itself. I'm not sure how small a chainring I could get on the existing cranks. Plus it has one of the cool old Raleigh chainrings (like this one), and it would be a shame to have to swap that out.

The Sturmey 5-speed looks promising, though. It's available with a 114mm OLD, so no spreading would be needed at all, and the range isn't bad. Plus direct drive is 3rd gear, which will work a lot better on a large-wheeled bike.

I know the SA 8-speed had some quality-control issues at one point. Does anyone know if the 5-speed had similar problems?

Addressing your three questions in reverse order!

My impression of the SA-8 is problematic not because of quality-control issues, but the difficulty of getting it adjusted just right. If the cable is off by a tiny bit, the gears skip occasionally (which is really annoying). It doesn't help that the most common use for this hub is folding bikes, and the process of folding can change the adjustment-- so this compounds the problem. SA has reportedly fixed this problem with the new, improved version of the hub (which is good news) which they won't sell until they sell the older ones (which strikes me as bad policy).

I believe the SA-5 is an old design --fifty years old, or so-- that has been improved over the years. It is essentially an expanded version of the AW hub, on the market since the 30's, which in turn was an improvement on even older ones. I have no first-hand experience with it, but I have not heard of any problems with adjustment, quality control, or other issues.

Finally, you're absolutely right, it will be difficult to find a cottered crank with a chain ring small enough to use the SA-8 hub with a 26" wheel. You can get a decent one with a 36T chain ring (I have one, and it's fine), but that's not small enough for your purposes. Finding a cotterless spindle that will fit your existing BB won't be THAT difficult; but a cotterless crank, with a tiny chain ring, would look terrible on a Superbe.

My vote: SA-5 hub. Or spread the frame and use a Nexus-8.