It depends on what you're looking for as far as amenities. Things start closing down before your planned dates including some of the lodges so that might narrow the field as well. We can have real nice weather in October or, like October 2013, one of the worst blizzards in the state history.
This time of year is heading into the off season. We don't typically recommend staying in Rapid City, but if you do get bad weather like Road Runner mentioned, there are more services that you can utilize. Unfortunately, nasty weather can occur. Also keep in mind some attractions will be closed or operating on more limited hours.
We've stayed at Blue Bell Lodge inside CSP twice and have been very satisfied with the cabin. However, as mentioned above at the time of year you will be visiting some things will already be closed/closing for the winter. On our first stay at Blue Bell, we had booked a stay for one night, two nights before they shut down for the season. As we checked in, we were shown the menu for the restaurant, with many things already crossed off--once they were gone, they were gone--no more food arriving since they were closing. In the time it took us to get to our cabin, wash up and get back to the restaurant for dinner, a couple of additional items were gone. We still made it work, but do be aware of that possibility.
Perhaps Keystone might make a better base for your family if you are looking for commercial activities for the kids--although check on those to see if they will be open at that time of year.