What is the source for this practice?
This article suggests
Perhaps the earliest mention of beer at a shalom zachor is by Rabbi Yaakov Halevi Lifshitz (1838–1921) of Kaunas (Kovno), Lithuania, in his work Zichron Yaakov, in which he chronicles the life and history of Lithuanian Jewry. He describes how they would proclaim in the synagogue after Friday night services that “so and so” invites the congregation to a shalom zachor, and how at the shalom zachor they would serve different types of beans and beer, or a different “social drink” (chamar medinah).
There is room to speculate that we drink beer either because it is made from barley, which is round (like beans), or perhaps because at the shalom zachor, there is a custom that lighter refreshments (as opposed to a full-fledged meal) are served.11 Thus, we serve beer, which is considered less formal than wine.
You can read the Zichron Yaakov (1:22) inside here.
There is an elderly Jew with lots of knowledge of tradition that I knew a while back. He came to the Shalom Zachor of my oldest son. There he said the reason for the custom is that Chickpeas were traditionally made with lots of pepper and were very sharp. Beer was served to cut the sharpness of the chickpeas.
I can attest to the sharpness aspect as I have occasionally been served the peppery chickpeas (not the bland American stuff that is popular today) at Shalom Zachors and they are indeed spicy. He was saying that this is the way chickpeas were served more commonly.