Hi Lilliput
The problem with most of the Med - Greece, Spain, Croatia etc. - that time of year is that it can be lovely but it can be very changeable at that time of year as well so you might find it a bit wet and windy. We've experienced both types of weather in the same area of southern Spain in mid October.
If you go a bit further south, the good weather is much more guaranteed, so I'm seconding both Lanzarote and the Red Sea. I think Cyprus could also work but have never been myself.
Lanzarote absolutely does not deserve its Lanzagrotty reputation! It's always a pleasant temperature, there are some absolutely gorgeous sandy beaches for you and the kids, the volcanic landscape is stunning and fun to see, and it's really quick and easy to drive all over the island to visit different resorts, beaches etc. For a bit more activity, there are some lovely easy walks you can do as a family e.g. along the beach at Famara which has amazing cliffs. I think the nicest resort is probably Playa Blanca in the south and this is near some gorgeous beaches such as Papagayo.
I also second the Red Sea in Egypt. Sharm el Sheikh is a nice resort - purpose built, so doesn't really have much character, but done very nicely with lots of pedestrianised areas etc. You can get some great deals to lovely hotels there, and there is loads to do for the whole family. If you all like swimming then the snorkelling is truly amazing and even though loads of people go there to go diving, you can see as much sealife just by snorkelling (I can vouch for this as I was stuck just snorkelling at Christmas due to being 6 months pregnant and saw more stuff than DH who was diving!) You don't have to go on boat trips to do this - just stick your head under the water from the beach and you will see truly amazing things! - so it's something all the family can do. You can also do activities in the desert which your kids would probably enjoy such as camel rides.
Hope that helps!