The street musicians featured in HBO's hit TV series Treme make the best money playing the high-traffic tourist areas, such as Jackson Square and Royal Street (during the day). The real estate around Jackson Square is too valuable to just hang out, and those just sitting around will be nudged aside in favor of folks who want to make money. When the buskers want some down-time, they often congregate by Washington Square in Faubourg Marigny.
Washington Square is a nice public park in the city's second-oldest neighborhood. Faubourg Marigny is the area just to the east (downriver) from the French Quarter. It's named after Bernard Mandeville de Marigny, who owned the plantation that sat on what is now the faubourg. In 1806, Bernard subdivided the plantation and sold lots in the area. Over the years, "the Marigny" has become a residential extension of the French Quarter. As the Quarter becomes more commercial and tourist-y, those who want to live there migrate more and more to the Marigny.
That's what you see with the street musicians in the show. Annie and Sonny often play in the 500-600 blocks of Frenchmen, which attracts a bit of a crowd in the evenings. There are a number of restaurants and clubs in the Marigny, so it's all logical.
Washington Square as a hangout is also logical, which is why Annie and Harley Watts (played by Steve Earle) are out along the fence. Annie watches as Steve works on a song he's writing, and ends up contributing some lyrics.
Originally named "Founders Park," Washington Square takes its name from the first President, but indirectly. It's actually named in honor of the Washington Artillery, which is now a unit of the Louisiana National Guard. The Washington Artillery (141st Artillery) has seen action in every war the US has fought in, from the Mexican War in 1845 to the Iraq war of 2003.
The photo above is a shot of the square from the late 1800s by George Mugnier.