I don't know much about what they do today, but your first thought was correct, they date to the late 19th century wave of Jewish immigration. Needless to say there's no pressing need to assimilate Jewish immigrants today, so they've probably found something else to justify their existance, but they've been around for a while. If you make horseshoes or buggy whips, you go out of business when you're no longer viable, if you're a social organization you simply redefine yourself.
The Hebrew Immigrant Aid Society is indeed over 100 years old, founded in an era where they did important and necessary work in helping Jewish immigrants migrate from Europe to the US and get assimilated to their new land. They were active as well when an acute need arose for European Jews during the Nazi era and then post WWII. They continued along that path for Soviet and other Eastern European Jewish immigrants to the US and Israel in the last few decades of the 20th century.
But now, undoubtedly because the need for their services for Jewish immigrants has waned, they need something else to do. So they have internationalized their reach to offer aid to all immigrants and refugees, apparently including illegals in the US, where they rail against law enforcement action and instead boast about promoting their idea of "immigration reform," which appears to be nothing short of amnesty.
What's behind their program, in their perverse logic, is that they are applying "Jewish values" to the American immigration problem. And they are using occurence of the upcoming festival of Passover to exploit their push for what appears to be something possibly even more accomidating to illegals than the defeated McCain-led immigration bill of 2007 entailed. How does Passover and "Jewish values" come into play here? Believe it or not, they analogize the "oppressive" US government today to Egypt under the Pharoahs and the illigal aliens in the US to the enslaved Jews of ancient Egypt - strangers in a new land! Never mind that the Jews of ancient Egypt were sold into slavery there and today's illegal aliens in the US willfully come into the US against our laws. Or that the Jews of Egypt had no employment opportunities other than serve the regime. But this is how devoid of reasoning the Jewish Left and those that run some of these organizations nowadays are.