Where to stay before late morning flight from Stansted

When I had early flights (8 AM) I would stay near the airport. The stansted express gets you from London Liverpool St station to the airport in just under an hour. If you are staying near the Liverpool St station you could stay in London and take the 8:25 train and be at Stansted about 2 hours before your flight.

I am risk averse, and would fret about there being an incident on the train line from London.
There are several hotels near the Airport-
https://www.stanstedairport.com/flights-and-hotels/hotels/stansted-airport-hotels/

I would also look at the nearby town of Bishop's Stortford for somewhere to stay.

I would always look to arrive at a UK airport 3 hours beforehand. It may be 2 hours for check in etc, but allow the extra hour for any problems, whether on transit or driving/taxi.
That is what any train or coach operator will advise.

Okay, that's good info. I wasn't planning necessarily to be in London but instead was thinking of Cambridge (will be coming from the North), which is apparently 30 minutes by train, running once an hour. Is that too far out?

London Hotels near Liverpool were crazy expensive for the night I looked at, and we don't plan to be in London anyway ($400 - $800 for one night for 3 adults)

Cambridge Station is a bit out of the centre. It's easy enough to get in and back out the previous day, but in terms of ease of departure I would stay at the Ibis Cambridge Central Station which is exactly where it's name suggests.
For the previous day the city centre bus- very frequent- leaves from straight outside, but you can walk it in 15 minutes or so.
I agree, far more budget friendly.
Then aim to get the 0735 train to Stansted next day- Advance fares currently available. If that train had a problem you have the comfort of the next train being at 0814- arrives 0841.

which is apparently 30 minutes by train, running once an hour. Is that too far out?

not too far out but what will be your Plan B if something with a train or the line causes delays or cancellation?

East Anglia (the area, not the train company in this case) is not exactly renowned for rail perfection, and Stansted from Cambridge is at the very end of a long route so you would be at the mercy of things back up the route. (I used work those trains before they were given to Cross Country).

An hourly train isn't very frequent or resilient.

The back up plan is National Express from the City Centre at 0800, arrives at 0850, also an earlier journey at 0545. As it happens the 0735 train originates in Cambridge, although the 0814 has come from Birmingham.
But from Bishop's Stortford there is a frequent all night bus service as well as Stansted Express trains.

If you look on Booking.com there are a number of places in Bishop's Stortford, Stansted Mountfitchet and at the Airport itself.
Two rooms at the PI at the Airport that night come to a total of £164, and they have a 24/7 private shuttle bus to the airport-
https://www.premierinn.com/gb/en/hotels/england/essex/stansted-airport/london-stansted-airport.html?ARRdd=02&ARRmm=05&ARRyyyy=2023&NIGHTS=1&ROOMS=2&ADULT1=2&CHILD1=0&COT1=0&INTTYP1=DB&ADULT2=1&CHILD2=0&COT2=0&INTTYP2=DB&BRAND=PI

Thank you to everyone who thew in their comments here. I read everything, and reconsidered Cambridge.
This is one of several reasons we decided to not leave from Cambridge, but instead from London (but not the $400-$800 a night places close to Liverpool St Station :-O )

We will be traveling with our daughter, who has also gone from London to Stansted, so I'm not concerned.

Thank you to everyone for spelling out the logistics of this small journey. ;-)