What I often do is create a media folder inside the project folder on my machine where the Hype doc resides and place all the images in the folder. I make sure they are of an appropriate size (placing an oversized image can cause problems, so it is important that the images are no larger than necessary) and optimized (open them in Photoshop and select File > Save For Web. Sometimes I use a dedicated optimization plug-in, such as TinyJPG/TinyPNG). When I output the file I end up with everything in the same folder - Hype doc, media, project resource folder and html doc.
If your Hype doc can't find the images, you can select the individual images in the Resources inspector and re-link them to the corresponding images in the media folder.
If you are having issues maintaining links on your local machine, it is possible that your directories might have some corruption. In this case, you might want to grab a copy of DiskWarrior and have it scan your directories. If it notices any discrepancies, this might be the cause of the problem. Make sure your drive is backed up before you do anything. Then run DiskWarrior and have it update your directories. This will make sure that the links are good. If the linked files aren't showing up in Hype, this might be because of the damaged directory. If so, re-linking them at this point would resolve the issue.