Suggestions for things to do around Bourton-on-the-Water

Excluding the obvious of visiting any number of villages, there are the Neolithic/Bronze age Rollright Stone or Belas Knap Long Barrow, Blenheim Palace, Malmesbury Abbey, or Hereford Cathedral. Oxford is close by. It really depends on what you want to see and do.

Hailies Abbey
Gordon Russell Design Museum in Broadway
Hike to the Broadway Tower
Sudley Castle
Snowshill Arms and a wander about the church and graveyard across the road

If you do decide on Hailes Abbey, also go into the church next to it which has some lovely medieval wall paintings.

Alternatively go for a ride behind a steam loco on the Gloucestershire Warwickshire Railway

We really enjoyed Blenheim, so much so that we visited twice. On the second day we parked at the Butterfly House and walked over to Winston's grave and memorial at the family plot at St. Martin's church across the street in the village
Upper and Lower Slaughter are both quite pretty but are so small that they can get crowded in a hurry. Best to park at Upper Slaughter and walk down to the lower town along a very pretty little stream. It's one place we wished we had spent more time in the area.
Broadway is also a pretty little village, and there's a pleasant hike ( maybe 1.5 miles) to a feature called the Broadway Tower that's worth doing.

Thank you everybody for your feedback. It is really appreciated.

We were in the area in May, you will love it!

When you say you have a hire car, will you be the driver or are you hiring a driver also? It makes a difference in terms of parking and how long you would want your day would be.

Having been there in 2018, I didn't care for it but went with a friend who I dusted and was driving. I insisted on going to Bampton. I liked it better.

In case you're not already aware, the village of Bampton that Sandra mentions above was a principal Downton Abbey film location. If you're a fan of the series you'll recognize many of the landmarks immediately. Plus, it's just a pretty little place in its own right.
One caution: get there early before the daily line of tour buses starts to roll in - as you can imagine it's become a popular pilgrimage site for fans of the program.

I second the suggestion of walking from Upper Slaughter to Lower Slaughter. It will be best to do this on a weekday, if possible, to avoid the crowds. That walk is one of our favorite travel memories of all time.