Lee, in speaking of a altering a concrete drive, plus doing so cheaply and as a DIY project, could be a tough order to fill.
What about how the water gets there, Lee, AND...what keeps it there?
Could you not do some grade correction to the yard which would divert water from heading to the drive to start with, or which would allow the drive to drain to the yard and be carried away with a swale?
Lee, another possibility would be to put in a small french drain that would run alongside the drive. If this is a possibility, I'd suggest a perforated pipe in a gravel collection point, but switch to solid pipe for any long run beside the drive transporting the water away; free-flowing water next to the slab could present new problems.