My most pertinent question would be: How computer savvy are you? Second would be, how much time do you intend to spend away from power outlets?
The Portable Media Player is going to have the longest battery life and store the most content with the least amount of fuss. You can put dozens of movies/shows on them and don't need to carry around any DVDs at all. Battery life should be easily double the 3.5 hour best you'd get from a DVD player. Downsides are that you must 'rip' and re-encode any content you want on the device. This is time consuming and can be somewhat complex depending on your skill level. EVERY movie/DVD needs this done, but once it's done you can load it onto multiple PMPs so if you have more than one kid, each will be happy.
The PortaDVD is the easiest and lowest cost option. Most have auto/air adapters available so those and a carrycase full of DVDs are pretty much no-brainers when you travel. Downsides are size and fragility. DVDs are not easy to damage but it happens. Same with the portaDVD player. It's unlikely but can happen. Not such a big deal now that they're very inexpensive but it's a concern. They'll have longer to much longer battery life than an equivalent laptop and are very easy to use.
The laptop is the most flexibile option and combines the benefits of the PMP and portable DVD player in that you can rip and transcode movies to the hard drive OR play DVDs in the drive. Movies from the hard drive will make the battery last longer but at best you're looking at 3 hours maximum and that's with a fairly modern laptop. Upsides are that you have full laptop and WiFi functionality. You can use it as a storage repository for digital photos, etc. Downsides? It's still a laptop, it's still more fragile than the other two and an HDD crash will mean you're dead in the water. Unlikely but a potential problem.
A small one that's workable will be $300-800 depending on whether you go used or new. The Dell 710m can be had refurbished with the extended battery for $750 or so. Averatec's lowest end model is $699 but for the price difference I'd go Dell. Otherwise, used you'll be on Ebay and that's a crapshoot.
My recommendation? Unless you would carry a laptop normally, want one, or already have an extra, go with either the portable media player or the portable DVD.