Hi! Sorry I meant to respond to this and I can see someone has beaten me to it with a very helpful response!
Kotor Aquarium (small but something to pass the time). Cats of Kotor. Just wandering around Kotor and taking it all in! There's not loads of specific child activities but of course the beaches make up for it.
Good beaches we tend to go to but these are outside the bay (but a short drive):
Plavi Horizonti (I think this will be the closest sandy beach for you)
Movida beach
Rambla beach Lustica bay
To be honest when you're driving around the bay you will find plenty of small spots you can pull over and enjoy a swim. Look at Orahovac too (couple of restaurants and more of a local beach but beautiful).
Other things but a drive away:
Lovcen cable car (just opened in July)
Ostrog monastery
Lake Skadar
Lovcen National park (even in summer it's nice and windy)
The town of tivat, visit Porto montenegro for the evening
Lustica Bay
Porto Novi
The last 3 are all "luxury" resorts but fun to walk around and often have something on. Google Lustica Bay Kids Club, there are activities on all day and I think it's around 5e an hour but it keeps your kids entertained and you can have a coffee!! They have a really active Instagram page.
Agree with the person before saying to hire a boat, soo much more of an easier way to see the bay. Anyway, hope this is somewhat helpful!
The main thing I would say is be prepared for the heat and the fact you may end up being inside from 11-3 everyday as the sun is very strong and intense!! Enjoy your trip!