I think 800 to 1k is the sweet spot. You get nice components, a somewhat light frame and it has the chance of lasting you a very long time. One upgrade I can highly recommend is getting a Brooks B17 saddle, they make riding so much more comfortable I would never ride a bicycle without one again. Personally I found great luck with a 300€ singlespeed that I ride daily for four or five years now, but singlespeeds aren't for eveyone and the lack of expensive shifting is the only reason mine was this cheap.
I'm an everyday single speed rider. I like it for everything except very steep hills of course, but have zero problems otherwise. Iinstead the simplicity and lack of maintenance are great, it's much lighter than any other bike for 300€ and I really can't complain. I like my freewheel though, Fixed has lots of problems in the city if you ask me.
I upgraded to a Brooks saddle because the stock wasn't that great and since then it only brings smiles to my face.
Just FYI Brooks saddles are on sale on Amazon. $65 for black with black rails and $66 for honey. Linky
I can highly recommend getting a better saddle, that is certainly worth the money. I love my Brooks B17 and it should fit most people and bikes. That being said if you already struggle with fit better try the saddle in a store if you can.
If your fingers go numb it's likely your handlebars are too low / too far out for you, making you lean on them to much. You could try a shorter stem (make sure the diameters are correct) and see if it gets better. It's not much of an investment and can make a hell of a difference.
I will never ride anything other than my Brooks B 17 again, it's just as good as everyone says it is. I prefer it to the spring models, less squishy and I found it astonishingly comfortable from day one. I have mine on my singlespeed road bike and find it works fine for a mix of relaxed and sporty seating position like I'd expect you to use while hauling a trailer. I'd say the 88$ Amazon calls up for the black model would be well worth it considering the comfort and longevity factor. I have also made good experiences with this Selle Royal if you want half that price for a pretty good saddle.
In the UK they are £55 on Amazon. Within OPs budget
Good multi tool would be cool and would get used.
This is a better looking gift which might go over very well or not depending on their scranus.