In the library mode go to the file name and click on the symbol to the right of it, then use batch rename to rename your files.
If you don't want to change the file name itself, then use the title and or caption field in the meta data panel to give names to your images and set this to show in the web gallery instead of the file name itself.
You can't rename the actual files from Web as they export. There's nothing in the SDK on it either.
Lightroom does change the naming to make it websage. For example, spaces are turned into underscores.
If you need sequential files, I'd suggest getting the files in order, then exporting the files as highest quality Jpeg with a sequential naming system.
Click the Add to Catalog checkbox, and perhaps use 'The same folder' and then use a subfolder for organistion.
Use these new images to create the web gallery.