Some humanitarian groups were founded specifically to challenge the travel restrictions. In the late 1960s, idealistic Americans, mostly without the required papers, traveled with the Venceremos Brigade to help in the harvest of sugar cane.
Another sugar-related and nonreligious group, the M.S. Hershey Foundation, brings medical supplies to the Sisters of Charity of St. Vincent de Paul in Havana. In the early 1900s, chocolate patriarch Milton Hershey built a sugar refinery and a town in Cuba named, naturally, Central Hershey. The foundation leads a tour that visits Ernest Hemingway's house but also takes tourists to the Valley of the Sugar Mills to see restored plantations. Some would argue that reforming sugar plantation conditions was one of the goals of the Cuban Revolution.
There's one Cuban aid project with a particular charm and special meaning for an island with many musicians. Piano tuner Benjamin Treuhaft has shipped a couple of hundred donated pianos to Cuba. He takes teams of tuners to fix those and also Cuban pianists' instruments that have gone off-key from tropical humidity and wood-chewing insects. Treuhaft, whose parents famously challenged authority, began his charity to flout the trading ban. (Treuhaft's mother, Jessica Mitford, was the daughter of an English peer, a Communist, and the author of The American Way of Death. Her husband, Robert Treuhaft, was a California labor lawyer and civil rights activist.) Treuhaft has been issued a Commerce Department license allowing the export of the pianos to Cuba. He says one of the conditions is that the pianos are "not to be used for re-export or the purposes of torture."
If you're looking for education about Cuba, the San Francisco-based group Global Exchange has a wide array of themed Cuban trips, with tours centered around music, the environment, Afro-Cuban culture, and medicine. The group is a human rights organization rather than strictly humanitarian. You'll be lectured on the folly of America's Cuba policy.
Whether organized into tour groups or not, most travelers bring large supplies of over-the-counter pills to give to Cubans. By all accounts, the country has a good system of medical care but serious shortages of medication.
Whatever the auspices of your Cuban trip, don't underestimate the value of just going and talking to people. Human welfare, you may argue, is not advanced by Americans drinking mojitos and dancing to Afro-Cuban music. Maybe the world is not a measurably better place because you go to see the island's bee hummingbird (the smallest living bird) and other natural wonders, fortuitously preserved by the fact that the Cuban economy has not exactly been a powerhouse.
Humanitarianism is also kindness, sympathy to other human beings. You'll be doing a service by keeping your eyes open and bringing back an accurate picture of what life there is like. Cubans are, in general, markedly warmly disposed to Americans. (The two places in the world where Americans are most appreciated, in my experience, are Havana and Hanoi.) You'll be particularly warmly received if you can talk about chess, baseball, or even Desi Arnaz.