It can be argued either way to stay at Heathrow or in central London.
If staying at Heathrow stay at a hotel by Terminal 3. The first tubes are at around 6am, so aim to be on a very early tube (in fact being a Friday to Saturday there is the all night 24 hour tube from Heathrow into central London).
You could get off the tube at Westminster. Even (maybe especially in the dark/dawn) seeing Westminster and Whitehall is special (I am an inveterate really early arriver into London used to be being there at 6 to 7 am. Walk up Whitehall to Trafalgar Square and take the bus #15 (destination Blackwall) down Fleet Street past St Paul's Cathedral to the Tower of London/Tower Bridge. Journey time that early on a Saturday 15 to 20 minutes.
Then the first Thames Clipper/Uber boat westbound is at 0844 to the Embankment or 0903 to Westminster (arrive 0903 and 0927 respectively) then from either pier take the tube back to Heathrow- you will be back by about 1000 or 1030 respectively. Plenty of time to check out and find your way to check in.
With a layover like that set your alarm clock early to take full advantage of the situation.
Having said all that there is engineering work on the Picc line showing for Kings Cross to Osterley (on the way to Heathrow) on the night tube/all weekend.
So if you find a hotel close to Paddington that night, then start your explore from Paddington on the Circle line to Westminster. Get that first boat back to Embankment, Bakerloo line back to Paddington, check out then Heathrow Express back (Earlybird £10 tickets are available for that date, cheaper and faster than Elizabeth Line). Same on the Friday night- Heathrow Express to Paddington on a £10 Early Bird ticket.