I don't know of any simply way just to add numbers to filenames in the way that you suggest (but see the last part of this post for a quick tip that might help you anyway). However you might find one of these programs useful:
http://download.cnet.com/File-Renamer-Basic/3000-2248_4-10306538.html
http://download.cnet.com/A-F-5-Rename-your-files/3000-2248_4-10135889.html
http://download.cnet.com/Rename-It/3000-2248_4-10028255.html
(NB: When installing software downloaded from CNET, take care to reject any pre-ticked offers to install 'extras' such as toolbars, etc. In particular, I note that the second and third links above say 'Installer Enabled'; when you see that it's vital to select the 'Custom installation' option, as that allows you to ditch the rubbish).
However this simple way of renaming batches of files might be good enough for you:
Go to the folder where the files are held.
Select the relevant files. (To select all of the files, press Ctrl and A together. To select a continuous block of files, click on the first one and then hold down Shift as you click on the last one. To make a selection of lots of individual files, click on the first one and then hold down Ctrl as you click on each additional file).
Right-click on the first file in the selection and select 'Rename'. Enter a name of your choice (such as 'Picnic') and hit Enter. The first file will then be renamed as 'Picnic', with the subsequent files being renamed to 'Picnic(1)', 'Picnic(2)' etc.