This is one where OP can help his mother draft an appropriate short and concise (needs to be 3-4 declarative sentences) email to UA asking UA to honor the $400 credit offer on the grounds that she simply did not understand the issue at hand.

While this is a she said/she said, UA may simply honor the request. If UA doesn't, there is little to be done here because it would mean that the agent didn't note her record and nobody will recall this after the passage of time.

I would focus on the relevant details:
1. Flight oversold
2. Mother offered $400 to VDB via FRA
3. Mother asked for M&M credit and told to check after-the-fact.
4. Mother understands that not happening here.
5. Mother asks for originally comitted $400.

Remember - UA can see the transactions in the PNR notes, so the rest really isn't helpful.