Is there any reason a consumer customer can't have a repo sweep account for his/her personal funds?

Why would they need one; they can have any type of interest-bearing account. Simply bump the rate up on an existing account product to accommodate your "special" customer if you are needing to pay them a higher interest rate (if your system will allow you to do so).

I asked why they would want it too, but the banker asked the question again. There is nothing that would prohibit it, is there?

Well, there may not be anything to prevent it. However, it's generally not a good idea to set up something like this for one customer. Disclosures for a consumer may be different than the ones you're used to providing a business. You may not have expertise in handling this kind of a sweep. Exceptions to your normal list of service offerings invite problems because they can often end up without the sorts of controls that you'd build into a full-fledged service offering, and there can be problems with "ownership" of the service when key personnel are away.

I would go back to the account officer to determine specifically what you need to accomplish. Once you know the goal, you will be in a better position to discuss options available.