Merely adding a name to an account doesn't necessarily mean the mother gave that money to the other person. The fact that she was the exclusive contributor, received all dividends, and paid all of the taxes, could easily mean it was an "incomplete gift". If that is the case, the Basis may get a full Step-Up. Estate planners often create "intentionally defective trusts" to exploit the full Step-Up Basis for an Incomplete Gift.
This is NOT my area of expertise, and it may depend on State laws, so the OP may want to consult a local Estate expert.
