Dear Sleuths of the 100 Hour Board,
Who is the drummer boy depicted on the drummer boy quarters?
I found a story in the April 1993 issue of the Friend ("Woman of the Dead", by Helen Hughes Vick) saying it was Gideon Burdick, but I couldn't find a more legitimate source than that. It's my great-great-great-great-great grandfather, so it's important to me!
-That Curious Cat
I have to tell you up front that this has been one of my favorite research projects yet.
I searched around the internet for a few hours looking for the answer to your question. As I was looking around, I came up with a few facts:
1. Gideon Burdick was a drummer boy in the Revolutionary War. It seems a fact that he was in Washington's army, and evidence suggests that he was even the General's personal drummer in the Battle of Trenton. According to this site, he was born in 1762, making him 14 years old at the time of the battle.
2. The bicentennial quarter (seen here) was designed by a man named Jack L. Ahr. The Mint issued an open contest to design the bicentennial coins. Mr. Ahr was the winner of this contest, as evidenced by this March 7, 1974, press release. In the opinion of this author, the drummer pictured on the quarter looks much older than the 14-year-old Burdick who served under General Washington.
3. Some people claim that the drummer on the back of the bicentennial coin is Gideon Burdick, but not many. One of the few sources I found suggesting this was the one you mentioned, a story from the LDS childrens' magazine <I>The Friend, entitled (rather grimly), "Woman of the Dead." The story tells about a young girl trying to research her ancestors, and how she finds information about her progenitor Gideon Burdick:
As she took the papers that Aunt Hattie handed to her, Drummer of 1776 jumped out at her. Rebecca’s heart started to pound as she read that Gideon Burdick had indeed been the fourteen-year-old drummer boy who accompanied General George Washington on his historic trip across the Delaware River on Christmas Night, 1776:I tried to get access to that article, but couldn't find it before this answer needed to post.
“The muffled sounds of Burdick’s drum encouraged the soldiers through the snow and sleet to the ice-bound Delaware River.
“Washington and his men won the struggle with the icy currents of the Delaware, defeated the enemy and turned the tide of the American Revolution. The drummer boy’s drum was no longer muffled as he beat out a signal of victory.
“Gideon Burdick’s memory has been immortalized with the issue of the new  25 cent piece by the U.S. Mint.”
I also found this blog, in which the author writes,
I want to tell you a story about a woman named Rebecca. She was raised in New York, and her father was a famous drummer. --Not a drummer in a rock band, but the drummer on the back of the bicentennial quarter. Gideon Burdick, as a young teenager, was George Washington’s drummer as he led troops across the Delaware in the Revolutionary War: “The muffled sounds of Burdick’s drum encouraged the soldiers through the snow and sleet to the icebound Delaware River."These articles are the only information I have that connects Burdick to the bicentennial quarter.
I was starting to get frustrated. I couldn't find sufficient evidence to say definitively that Burdick was the drummer on the coin. My efforts to find out if it was someone else specifically were similarly fruitless. It really bums me out to have to return an answer of, "Boy, I don't know!" to a Board reader, so I kept going.
Then I had a crazy thought.
I knew the man's name who designed the quarter. Maybe I could just go straight to the source!
Some research got me in contact with Mr. Jack Ahr of Illnois, who is just as pleasant a man as you've ever met. Mr. Ahr and I had the opportunity to talk on the phone for a while about the bicentennial quarter.
Mr. Ahr was encouraged to enter the national contest by his wife. He claimed he was too busy, but thanks to her loving goading, he eventually submitted the now-famous design featuring a Revolutionary drummer. He was thrilled to hear he won, and was rewarded with a trip to Washington, a tour of the White House, and even a military brass band playing in Chicago at the unveiling of the new coin. He spoke of the time proudly, and said that he and his family still reminisce fondly about the fun and fame the winning design brought them.
Here's the bad news for you, though. Not only is the drummer on the quarter not based specifically on Gideon Burdick (Mr. Ahr claims to have never even heard of the man), it's not really based on anyone. Mr. Ahr said that the idea of the drummer was inspired by the famous painting The Spirit of '76 which features two drummers. Neither of these drummers was a model for the quarter (which is easily told just by looking at the two); Mr. Ahr just liked the idea of a drummer and fashioned the design himself (before anyone asks, I can find no evidence that the drummer boy in The Spirit of '76 is Gideon Burdick either). While the person on the quarter is no one in particular, the posture and form of the drummer are based on Mr. Ahr's son, a drummer who helped his father nail down the specifics of how the sticks are held and other drum essentials.
The drummer on the back of the quarter, though not inspired by anyone in real life, is certainly not nameless. Mr. Ahr told me that while he was laboriously sculpting the relief that would become the mold for the quarter, his children, for no reason in particular, nicknamed the drummer in the design "Luther." Mr. Ahr told me that if someone was looking for him, his children would respond, "He's downstairs working on Luther."
So, Curious, while I'm sorry that the drummer isn't your ancestor, I'm really glad you asked this question. My gratitude goes out to Mr. Jack Ahr, whose helpful and enjoyable conversation made this question a blast to answer.