Martin Luther King day was officially declared a national day of service by the King Holiday and Service Act of 1994. But I have been able to find no discussion on the internet of how the idea originated, and I believe that is worthy of some documentation. I once heard that the King family had been encouraging people to observe the holiday in that way, but I can find no reference to that. Wikipedia credits Senator Harris Wofford and Representative John Lewis as authors of the bill, but give no background. A quick search of the NY Times archives turned up no coverage that would indicate where they got the idea. I am guessing Howell and Lewis wanted to formalize a practice that had already begun, but I would like to see some more detail like an interview with one of them or coverage from earlier of MLK-related service activities. Is there such documentation?
This is a partial answer based on some more research I did. A Google Books search for the phrase "a day on not a day off" led me to this transcript of a 1993 hearing on the Holiday and Service Act. Apparently the original law providing for the King holiday also established a commission to "assist in the celebration" of the holiday. That commission, headed by Coretta Scott King, was apparently due to sunset at some point, and the purpose of the bill was to extend its life.
The tone of the remarks suggest that treating the day as an opportunity for service was not that well established by that time. The slogan "remember, celebrate, act" had been around. In this context, "act" to me tends to suggest political action, but it absolutely can include other kinds of service as well. It is just not specifically about volunteerism.
Senator Wofford, in his remarks, used the expression "a day on, not a day off," and this was the earliest time that expression showed up in my Google Books search. According to his Wikipedia article, he was "a noted advocate of national service and volunteering", and had played a significant role in the creation of the Peace Corps. So it makes sense that he would push for this idea.
Coretta Scott King also spoke at the hearings. In my mind, when I posted the question, I was partly trying to disambiguate whether it was King's family, or Wofford and Lewis, who had the idea. But Wofford and Lewis were both friends of the Kings, so the development of the idea could have been more collaborative.