We will be in Mumbai just for one night and a day, plus some of next evening as we have a very late flight. What are your suggestions for things to see and do with this limited time?

It's been since 2009 when my son and I visited Mumbai, but I'll give you a summary of what we did, with the caveat being there may be some different sites since then.

In any case, we first went on a walking tour using the DK Eyewitness book. The buildings are basically British from the Victorian era. We also visited the historic train station and the museum. We went to the inside market, but I would advise against it, since we saw a cage full of canaries, with half of them dead.

We took a boat ride to Elephanta Island. The caves were moderately interesting, but the boat ride was enjoyable.

We had extra time at the end, so we went to a shrub sculpture garden across from a beach. I forgot the name, but it's quite well known. It was a nice change from looking at buildings.

I love Mumbai for the food experience it offers. It has some of the best Parsi food joints in India, the only other competitor being Pune.

While not for the faint of heart, Britannia is a good place to visit for its awesome Berry Pulao. Though going on a culinary expedition in India when you have less than two days might not be for everyone.

Laundry ghat - walk down steps. Someone will greet you at bottom - take tour
Slum tour
Walk by India gate
High tea at Taj Majal

Walk or drive by queens necklace (waterfront area)

You can go via Bandra Worli Sea Link up, Marine drive to Gateway of India.You can visit Bandra Band Stand, Juhu beach and ISKCON temple. Hill road and Linking road are good for shopping.

High tea at Taj Palace hotel on the waterfront. Lovely food, room and views.
We had same limited time there so took a tour with Grand Mumbai tour leader Pranav. Very good, nice air conditioned car, choice of sights, interesting details. Could not have seen as much as we did by ourselves in the time we had.