what did the cherokee contribute to modern day society?

Syringes, rootbeer, rubberized clothing, beef jerky, and many of the tenets of the United States Constitution. Food staples (corn, beans, potatoes, sweet potatoes, and squash).

The Cherokee and those they descended from were on this continent for thousands of years before the white man set foot on it. Stories, legends and culture were passed down orally, but Sequoya developed a form of Cheorkee writing which allowed that lore and information to be transferred from those that held it (Medicine men, or Shamans) such as A`yûn'inï or "Swimmer, Gatigwanasti, Gahuni, and Inâ'lï, or "Black Fox" , into English.

The Cherokee used herbs for healing. An astonishing number have proved to be valuable in modern pharmacology. Many modern drugs have active ingredients based on these herbs that were chemically synthesized. More than two hundred drugs are derived from their healing remedies. Beginning in 1993, the National Institute of Health recognized the importance and use of herbal medicine as a treatment for certain types of illnesses.

Jack C. Montgomery, a Cherokee from Oklahoma, and a First Lieutenant with the 45th Infantry Division Thunderbirds, on 22 February 1944, near Padiglione, Italy, Montgomery's rifle platoon was under fire by three echelons of enemy forces, when he single-handedly attacked all three positions, taking prisoners in the process. For this he won the Medal of Honor. Charles George, a Cherokee from North Carolina, and Private First Class in Korea when he was killed on 30 November 1952. During the battle, George threw himself upon a grenade and smothered it with his body. In doing so, he sacrificed his own life but saved the lives of his comrades. George was posthumously award the Medal of Honor in 1954.

Will Rogers aka "The Cherokee Kid" is an American legend. He’s famous for saying “I never met a man I didn’t like". The outpouring of national grief over Rogers's passing is generally regarded to be the greatest show of national mourning since the death of Lincoln.

For an better understanding of Cherokee healing, medicine, spirit and culture, see the links below.

Knowing anything is one thing. Knowing everything about the contributions of the Cherokees or the Cherokee Nation will probably take a life-time of scholarly study. This having been said, I suggest you start with looking up "Cherokee" in Wikipedia. Most of the article is not about their contributions, but about how their proud nation was mistreated by the whites and chased out of their native land by the whites in the 1800's. However, one part of the article says, "Other famous people of Cherokee ancestry include the actors Johnny Depp, Liv Tyler, Shannon Elizabeth, Burt Reynolds, James Garner, Chuck Norris, James Earl Jones, Lou Diamond Philips, David Carradine, Charisma Carpenter, Wes Studi, Stephanie Kramer and Christopher Judge ; the dancers Mark Ballas and Eve Torres ; the musicians Steven Tyler (Aerosmith), John Phillips (The Mamas and the Papas), Eartha Kitt, Miley Cyrus ; the actress and singer Cher ; the singer, musician and actor Elvis Presley ; the singers Rita Coolidge and Tiffany ; the country music singers Billy Ray Cyrus, Loretta Lynn and Crystal Gayle; the boxer Jack Dempsey ; the painter Robert Rauschenberg ; the activists Rosa Parks and John Leak Springston, and the writers Mitch Cullin and Alex Haley." Needless to say the people who are in this list contributed to the contemporary American culture significantly in varying ways and degrees, Although this list focuses primarily on the people of celebrity status, there are numerous less famous Cherokees who contributed to the contemporary American society in every field of endeavor. I don't know why you asked the question you asked, but I am glad you are interested in looking into the contribution by the Cherokees because they, like many other minorities in American society, played and continue to play, significant roles in shaping the society we live in.