Blender allows you to render an animation to images and include the frame number in each filename. To get more control over the saved filename you can use a python script to do the rendering.
Paste the following script into blender's text editor and adjust the list of filenames to what you want to use for each frame. Instead of using blender's render buttons you can run this script to save each frame using the filenames you want.
A disadvantage is that blender's window will not get updated while rendering this way. You may want to do the render from the command line to get some feedback. If you save the script inside the blend file and name the text block
myrender you can use -
blender -b myfile.blend --python-text "myrender"
# the folder to save the renders
outpath = '//renders/'
# filenames is a list containing frame number and filename
filenames = [
scn = bpy.context.scene
for r in filenames:
scn.render.filepath = os.path.join(outpath, r)