London Hostel for Adult and Older Son?

When are you travelling? Even hostels are over £150 this summer

Timing will have an impact. Is this a summer trip? Are you willing to risk having no air conditioning? That's not likely to be available in a hostel; even many budget hotels don't have it. You might well not need a/c, but on the other hand, there might be a heat wave.

Saturday night is often by far the costliest night of the week, so if you can avoid spending that night in London, you may save money.

Staying outside the center of town--which I define as roughly the area circumscribed by the Circle Line on the Underground--can be quite a bit cheaper. That guideline will probably apply to hostels as well as to hotels and apartments, but now you're farther away from quite a few sights you may well want to see, and the Underground fares for two people will add up.

Yes stay further out to save money. If you stay near the new Elizabeth line it’s now very quick to get into central London from places like Ealing or even further out. You’ll find budget chains like Premier Inn and Travelodge everywhere and they’re a good safe bet. Nicer than a hostel.

For example, Premier Inn Ealing Broadway has rooms around £100 on various dates this summer that I looked at and it’s 10 mins walk from the station.

OP you might look up YHA and see if that works for you -- https://www.yha.org.uk/

I've used them in other areas of England and in London. It's been quite a few years but I liked both YHA St Pancras-- https://www.yha.org.uk/hostel/yha-london-st-pancras -- and YHA St Paul's -- https://www.yha.org.uk/hostel/yha-london-st-pauls. St Pancras is close to St Pancras and Kings Cross train stations and the tube station. The area around St Paul's is just my personal favorite area of London. The private rooms were pleasant affordable options, but I've noticed the prices have gone up a bit and these days I've been using Locke aparthotels.

To answer the above, we're planning on traveling July 10. Our preferred budget is up to $160. I've found some hostels and other spots on airbnb for under that budget that are in town, so that's promising.
I'll have to check on whether the options have AC though.
And thanks for the links above.

try university rooms https://www.universityrooms.com/en-GB/city/london/home/

https://www.lsevacations.co.uk/Home.aspx - Bankside House is a great location

Kensington Gardens Hostel has rooms just under £100 around 10 July. It is owned by the German YMCA and is near Lancaster Gate.

I've recently stayed at both the YHA Earls Court and the Wombats hostel. I would happily book them again. (At the Wombats make sure to stay in one of the rooms with no windows on the main road.)

I second the YHA hostels as mentioned by VAP.
YHA St. Paul's location is excellent.
Near the sights, great transportation nearby.
The number 15 bus from Trafalgar Square to the Tower of London runs right by the front of St. Paul's Cathedral.
It runs in the opposite direction, also, from Tower of London to Trafalgar Square.

YHA London Oxford Street is actually not located on Oxford Street, just near it, at 14 Noel St.:
https://www.yha.org.uk/hostel/yha-london-oxford-street
Great location, walking distance to shopping, the theatre, Soho, Covent Garden, and more.
Laundry room downstairs......Washers and dryers are a luxury when traveling!
Unlimited continental breakfast here, clean rooms, and secure front door at night.

If you prefer the Kings Cross area, a YHA hostel located on Euston Road is roughly across the street from Kings Cross station, , basically a few minutes walk. It's decent , have checked it out a few times.

I always stay in a B&B in Kings Cross, no AC with them either...basically , don't expect it in a hostel or a B&B.