You don't mention the puppy's actual age. Is it fully vaccinated yet? Bringing along a copy of the pup's shot record is a very good idea, as well as making sure it's microchipped and that chip is registered to you with your cell phone #. Also, be sure his/her collar has your cell phone # so, heaven forbid, if the pup were lost anyone finding it can actually reach you. My 14-lb dog rides in a basket on my handlebars. His harness vest is tethered to the basket so he cannot jump or fall out. You will want to make sure that whatever your pup rides in has a tether as well, to keep it secure & safe while you're in traffic.
If your pup is less than 5-months old I strongly recommend that you do not walk it in areas with high dog traffic (designated dog potty areas in rest stops, dog parks, dog runs, etc.). Warm weather means parvo will be on the rise and puppies are especially susceptible to this very fatal disease. It's just a precaution, but a good one. Walking your pup off the walkways or just out of human-oriented areas will be safer than the dog-designated spots. We brought home a 9-week old golden retriever from OR to SoCal and managed just fine by avoiding the dog potty areas.
Be sure to pack toys & supplies for your pup, too! A first aid kit wouldn't be bad, either.
You will need to be ready to stop and walk or exercise the pup as often as it needs. Be aware that, depending on the pup's breed and the weather, you may have to stop and give it adequate shade/water/cooling down time. Dogs perspire via panting and via sweat glands near their foot pads so if it's really hot and humid your puppy will need some help staying cool.
It would be a huge help to you guys if you acclimate your pup to his trailer or whatever BEFORE the trip.
I'm not trying to dissuade you from taking the puppy, but if you're going to take him it'll be much more fun for all of you if you're prepared. My little Havanese will be going with us on the road and I'm looking forward to it!