I think so. Recently I imported older project files into a later version of Premiere Elements, and they worked fine. Just make sure you have all your media files on the other computer or it'll show you that "Media Offline" message you get if you delete a video currently used in a project file.
Use the Premiere Elements 8.0/8.0.1 Project Archiver Copy option to save your project. See File Menu/Project Archiver. That will save your project.prel and all the media that went into the project in a folder named "Copied....". Transfer that "Copied..." Folder to a USB Flash Drive.
Transfer the "Copied..." Folder to the second computer, using this USB Flash Drive. In the second computer, open the "Copied..." Folder, right click the project.prel, select Open With, then Premiere Elements 10.
1. Remember than once you edit the Premiere Elements 8.0/8.0.1 project in Premiere Elements 10, "by the book", you should not be able to edit it again in Premiere Elements 8.0/8.0.1. However, even if you can, I would advise you not to do that to avoid still more unexpected events.
2. Also a good idea to finish a project in the version in which it was created, but, if you must, the above is how it works for me. There may be instances where an effect or feature might not be available in another program.
Please update us on your progress.
I second what ATR says about not opening a project created in an older version of the program with a newer version of the program. It almost always leads to buggy behavior and, often, a completed corrupted project file.
I very much recommend you finished your version 8 project in version 8 before moving on to version 10. Or, at the very least, keep a backup of your version 8 project just in case version 10 destroys it when you open it.