A standard double room at the Kings Cross Premier Inn will be £232 (flexible rate) on June 8, 15, or 22., without breakfast. That gives you an idea of what you can expect to pay for a budget-friendly hotel in London on a Thursday night in June.
I suggest you choose your hotel location based on ease and cost of getting to the airport on Friday, rather than some nearby attraction you want to take in. If your hotel is near a tube station you can easily go to any of the popular attractions for the afternoon and evening.
You will get lots of opinions on this, but our standard practice for the night before our flight home is to stay close to Paddington Station and book the low-cost advance tickets (£5.50) on the Heathrow Express for our ride to Heathrow the next day. The HEx is the fastest and most comfortable way to Heathrow, and takes you right into T3T2 or T5 in 15 or 20 minutes.
These must be booked 90+ days in advance, and they are limited in number. People have been reporting they are already sold out for some dates in summer, but I checked random Fridays in June and they are still available.
https://www.heathrowexpress.com/tickets-and-offers/book-early-to-save-money#/
So start by looking at these for your departure date, and if you can get the cheap tix then consider booking the new-ish Pilgrm Hotel close by Paddington( yes it is spelled that way).
https://thepilgrm.com/
Their Kingbed rooms are £229 in June, or a smaller double room is £219, both fully-refundable flexible rates, without breakfast. (I would choose the larger one).
When you arrive at Kings Cross station, it is an easy 10-minute tube ride from there to Paddington. Check in and leave your luggage, and then head out to explore London. This is just a few blocks north of Hyde Park, and you might like to stretch you legs with a walk in the park after that long train ride. You can walk through the park to reach some museums like the Victoria and Albert or the wonderful science museum, or the area around Buckingham Palace (a bit further away). The museums are free ( unless things have changed recently) and of course walking around is too.