Two weeks in Croatia.

Sure, renting a car and driving would be easy. Prices will be higher (and more traffic on the roads) if you are going in high season (July and August), but otherwise, driving in Croatia is easy, and driving in Bosnia is almost as easy (some roads aren't quite as modern but I found driving through Bosnia in 2015 quite easy). Make sure any car you rent has an insurance card for crossing the border. Sometimes car companies charge you extra for this, but you need to ask in advance to drive the car out of the country (especially out of the EU, which Bosnia is not in).

Mostar is a nice town, but it's not very big. You might consider adding a few nights and detouring up to Sarajevo, which is bigger and more historically significant. (e.g. the assassination of Austrian Arch Duke Ferdinand took place there in 1914, sparking World War I).

If you aren't going to go to Sarajevo, you can just take buses from Dubrovnik to Mostar and Mostar to Split and skip renting the car. The car is handy to have if you wish to make stops in between. There is reliable bus service in this region. The drive from Mostar to Sarajevo along the river is very beautiful; the drive from Croatia in to Mostar isn't really that special.

For car rentals, try Auto Europe (which will show you the car companies) or EconomyCarRentals (perhaps cheaper). To rent directly from an agency, try Sixt. No need to rent at the airports, either, FYI - you can find rental offices in the towns and do a one-way rental.

We can't leave the SF,Ca before 8/10, so we'll arrive in Dubrovnik on 8/11. There's jet lag to consider, so I suspect 8/11 is going to be finding our lodging, walking a bit and going to eat dinner. Rick Steves's tours spend three nights in Dubrovnik, so we thought we would too. One day we can do the day trip to the Bay of Kotor if that seems like more time than we need in Dubrovnik proper. We then thought we'd go to Mostar (per Rick Steves's guidebook suggestion) on 8/14 . Overnight in Mostar sounded more fun than a day trip from Dubrovnik in order to keep driving time down to a little over two hours and get to see Mostar in the evening. On 8/16, we plan to drive to Split and stay there overnight, so we're there and ready for the week of island sailing from 8/17 to 8/24. That leaves one day of leeway. Do you think we have time to go from Dubrovnik to Sarajevo on 8/14, Sarajevo to Mostar on 8/15 and Mostar to Split on 8/16, or is this too little time? Is there a better itinerary you can recommend within these time constraints. There are 4 adults and one nine-year-old, five suitcases, five "carry-ons" and a guitar, and laptops—at least—so a car would better than a bus. Thanks for your input.

It might work if you can do two nights in Sarajevo instead of just one - and make Mostar a long day stop. On the upside, Mostar is small and can be seen easily in a few hours on a day stop. On the downside, Mostar is a popular tourist day trip from Dubrovnik, so you might find it very crowded during the day and less so at night.

If you want to visit Kotor from Dubrovnik, you might want pick up the rental car a day early and spend two nights in Dubrovnik and one in Montenegro. (I'd prefer Perast to Kotor; be prepared for sky high prices in August.) The highlight for me of my visit to Kotor was driving along the bay, with its breathtaking scenery; if you take a day tour, you'll be in a bus or a van. I enjoyed stopping a few times along the bay for pictures, and I would have felt rushed trying to do it in a single day. You could drive back to Mostar inland (via Trebinje) instead of going back to Croatia.