You are talking about digitising new objects, so I have assumed that the replacement polygons for each existing point don't already exist in another table that can be joined, nor can they be derived from the existing point object by an operation such as a buffer.
In MapInfo, make the layer that contains the objects to be replaced editable in a map window, then select one or more of them and choose Edit > Clear Map Objects Only. This will delete only the object, retaining the attribute record. Now to digitise a new object associated with this record, select the record from the browser window and digitise an object as you would normally. On completion of the digitising, the new object will be associated with the existing attribute record rather than a new empty record.