I'm not sure where the myth comes from, but it seems to be an argument not based on fact, or at least, it's an argument made in bad faith.

According to various sources, in 1787, the Paymaster-General of the Continental Army, John Pierce, Jr., estimated that 231,771 men served in the Continental Army, and a further 145,000 men served in the various state militias during the war.

However, as has been pointed out since, the figures for the Continental Army, at least, includes multiple enlistments. So in some instances, men are being counted multiple times in that figure, once for each time they enlisted.

According to the book The Continental Army by professor Robert K. Wright, Jr., written for the U.S. Army's Center for Military History, in September 1776, U.S. Congress estimated they had about 31,000 troops to call on between the Continental Army and the state militia troops that were active duty, with the militiamen outnumbering the Continental soldiers two-to-one. (See page 87.)

By mid-1779, the Continental Army had 25,000 active troops at their disposal. (See page 151.) By early 1780, the Continental Army had about 35,000 active troops in their various regiments. (See page 154.)

According to the National Archives, post-war, there were some 80,000 pension and land grant applications sent in at the federal level. This, of course, does not include all the men who served. Not everyone applied for a pension or land grant. For example, some men died early without any heirs who could or wanted to apply--women who got remarried could not apply for their earlier husbands' pension or land. Some men died during the course of the war and never had anyone who applied for their compensation. In the first decades after the war, only disabled soldiers could apply for pensions, and this only changed after many soldiers had died from unrelated causes. And, of course, many more records were lost over the years.

So while at any given point during the conflict, there were probably not many more than 35,000-70,000 troops available to fight at the Continental and militia level, different people served at different times. Modern estimates put the number of U.S. soldiers who served at the Continental Army level somewhere around 100,000, and among militia soldiers closer to Paymaster-General Pierce's estimate of about 150,000. It could be less, but even a low estimate wouldn't put the total of all men who served either at the federal or state level much lower than 200,000. It could also be higher than that 250,000 total.

But for the sake of argument, put the number at 200,000. There were an estimated 2.5 million white Americans living at that time (the number of black troops fighting on the U.S. side wasn't very high--it was higher on the Loyalist side, since the British were promising freedom in exchange for service).

On its surface, then, that low estimate would be 8% of the population. But, of course, half the population was women who did not serve in active duty military roles. So the number of men in the U.S. at the time was closer to 1.25 million. And a significant portion of these were under 16 years old and too young to serve (though some did break the rules), or over 60 and were not obligated to serve (though some at that end served, too). There were also the infirm, due to diseases, or injuries prior to the war from the day-to-day risks of 18th Century life--as well as some who had gained those injuries from prior service during the French and Indian War.

Being generous with the age 16-60 population, say it accounted for 2/3 of all living U.S. people at the time. So 2/3 of that 1.25 million figure of U.S. men would put the number at about 833,333 men who were fighting age at that time.

That means that, roughly, 25% of all fighting-age men (200,000 / 833,333) served in an active military role on the Patriot side at some point or another during the war.

And, of course, that is only the Patriot side. On the Loyalist side, the numbers vary, but it is generally though that some 60,000 to 120,000 U.S. men served the Loyalist cause. According the the Canadian national archives, an estimated 5,985 Loyalist soldiers are named on the Loyalist lists that came to that country post-war. And only a fraction of Loyalists left the U.S., and of those, Canada was only one of several destinations (Great Britain and the British holdings in the Caribbean being the others).

So if you factor in those men, and give it a conservative estimate of 75,000 troops, then some 275,000 white men living in the United States served on either side of the Revolutionary War in a military capacity. Out of ~830K fighting-aged men, that's roughly 1 out of every 3 that served.

And this doesn't include all the people who supported the war effort in a variety of other ways. Among the support that non soldiers engaged in included: providing crops and other supplies at a discount or for free; offering transportation services; offering other free services to troops, anything from blacksmith services to spying; lending land to regiments passing through town for soldiers to camp on, instances of which there are countless stories; aiding in building facilities, including everything from guard towers to flour mills; offering medical aid for wounded soldiers; and even assisting in burying bodies. The actual military soldiers were only one piece of the pie who served the war effort in some way. Whole towns were engaged in the war one way or the other, particularly in those areas close to the war front. And the war front shifted throughout the war, so many parts of the country were touched by it at various times. While it's an impossible number to accurately estimate, if you were to include all the support services, you would come to a significantly higher number than 1 out of every 3 people who aided in the war effort on one side or another.

So where would you get a 3% figure from? Probably arguing in bad faith, along the lines of 80,000 pension and land grant applications divided by the total population of 2.5 million, which will give you a figure of 3.2%. But of course, this doesn't account for the fact that babies and the bedridden can't fight, women weren't expected to fight and didn't, there were Americans fighting on both sides of the conflict, the 80,000 pension and land grant applications is not a complete account of all the men who did serve actively in the military, and hundreds of thousands if not millions of people aided the war effort in non-active military capacities.

But I am not familiar enough with the 3%er myth to say for sure how they come to that conclusion. It's not one that any academic source would accept, though. Based on the fact alone that half the population were women who were not subject to serving militarily at that time, though countless women aided the war effort in the local capacities that they could serve, it shows that the 3% argument is not one being made in good faith. A similar argument would be that "only 10% of Americans were interested in defeating the Nazis during WWII" in a bad faith argument comparing the number of active duty troops to the total U.S. population at the time, completely ignoring all the people too young or too old or too female to serve, as well as all the ways in which non-combat personnel aided the war effort at every level of society.