There must be a name for this but I don't know it. It's a common problem when merging data sources.

If you have a good amount of data on existing mappings, you could learn to predict that mapping for each input field. The simplest thing that comes to mind is to use character ngrams of the source field name and predict the correct target field name (or predict that there's no match).

If you also have a sample of data from the customer, you could use properties of the data in each field as input as well -- the data type, range of numeric values, ngrams for string fields, string length properties, etc.

As for the business problem, even with automated mapping you probably need to force customers to review and correct the mappings or else you might end up with complaints from customers that didn't review.

All this isn't quite by area of expertise, hope this helps!