Based on some really quick math (300 unpaid interns, $10.10/hour each, an annual White House budget that is approximately $1.4 billion per year), it would add 0.43% to the White House budget to pay every single one of its interns minimum wage for two thousand hours per year. This is less than a third of what the Obamas spend on their vacations and is a few thousand dollars less than what they spend on maintaining the lawn and garden.
Now, I worked for the government as an intern who was not paid minimum wage (USCCR, 2L summer). It was reasonable, given the opportunity, length of time, and how enjoyable the work was. I was not supporting a family.
Note that all of those justifications also apply to a lower minimum wage: if a young person can reasonably decide that $0.00/hour is a reasonable wage, given the opportunity and their own financial needs, then they can also reasonably determine that $7/hour is an acceptable wage. But if people cannot negotiate for a wage that is lower than $10.10 an hour, that rationale should extend to unpaid internships as well.