Hi there Starfish and welcome to the Q&A Forum!
Are you familiar with EXIF data, photo information that is stored within digital images including date and time taken? If not, click on the link at the end of this post.
Check the Easy Share software help files to see if it reads exif data. You can also download free software such as Picasa which organizes photos and lets you read the EXIF data. It also has a timeline so folders and image files are automatically downloaded and sorted by date (It also organizes photos already on your hard drive). I use Picasa and have recommended it to many others.
As to changing the camera filename, dig around a bit more in the EasyShare menus and help files. There is probably place, other than in the Properties menu, where you can assign a different file name to photos being downloaded.
I've owned, or own, cameras made by Canon, Panasonic, Fujifilm and Nikon, and almost all programs allow me to adapt filenames.
I happen to be a big fan of Canon's ZoomBrowser software, and use it almost exclusively for downloading jpeg images from all my cameras.
Check the links below. If you need additional information, please let us know.
Related reading:
Exif datahttp://www.digicamhelp.com/learn/glossary/exif.phpPicasahttp://www.digicamhelp.com/processing-photos/basic-editing/picasa.php