We are looking into driving one way from Sarajevo to Dubrovnik via Mostar. Any problems with border crossings? Or are we better off not driving ourselves?
I drove a rental car last year from Plitvice Lakes (in Croatia) to Sarajevo, then Mostar, and then back into Croatia (to Split).
No problem whatsoever crossing the border in either direction. It was quicker than crossing between Croatia and Slovenia (which I also did in both directions on the same trip).
There shouldn't be any problems crossing the border, but just know that you will cross the border three times if you follow the Neretva River to the coast. Once entering Croatia, you'll be forced to enter Bosnia again at Neum, and then leave again a few kilometers later. None of the crossings should take very long and the scenery of the river and then the Croatian coast is fantastic.
"driving one way from Sarajevo to Dubrovnik". I will add the standard caveat that what you are proposing is a one-way international car hire. These usually incur large surcharges. Are you using a hire car, have you got a quote yet for this car hire?
You don't say when. Summer traffic has increased, and Bosnians tend to come home in summers. I would personally take the bus, and sit above the traffic with a good view of the Neretva. There was the train to Ploce (been there, did that) but I think now only to Mostar. There are also minicab taxis you can share, usually station wagons or minivans, between Sarajevo and Mostar, at least. Save the cost and complications of rental cars by taking transit of any kind, and enjoy the scenery in safety and meet a few people.
What about a one-day care hire entirely in Crotia, e.g., pick-up in Dubrovnik, drop-off in Zagreb? Do you think a drop-off fee would be charged?
The fees referred to are ones where you pick up the car in one country and drop it in another. Renting in Dubrovnik and dropping in Zagreb shouldn't result in any extra fee. The fee is not assessed in cases where you drive into other countries as long as you return to the original country to drop it. You should still double-check with the rental company that the car is allowed to enter Bosnia or any other country in your itinerary.
Heck, I drove from Mljet to Mostar one day and never even saw a border crossing. Discussing our drive north (same road as from Dubrovnik) the guy at my hotel told me, "When you see your first traffic signal, turn right". We did and didn't see another signal until we got to Mostar. Kept expecting some kind of a check but there were none. Plus, the drive was pleasantly scenic and completely deserted.
I think you must have been distracted or dozing. The first signal you come to after leaving Mljet on the way to Mostar is in the centre of Neum. Neum is in BiH, and there is no way (legally) to get there without going through the border crossing.