Professor Eulalie “Lally” Hicks was an American witch and a professor at Ilvermorny School of Witchcraft and Wizardry in the late 1920s. In 1927, Nicolas Flamel used his gold, phoenix embossed, magical book to have a conversation with Dumbledore but his page was blank so he flipped to Eulalie Hicks’ page. She urged him to help Newt Scamander and told Flamel that “we believe in you”. [CG]
"Eulalie" is the French form of "Eulalia", which is derived from Greek ‘eulalos’ meaning "sweetly-speaking". The surname "Hicks" is an English patronymic derived from "Hick", which is from the medieval personal name "Hicke", a pet form of "Richard". It is a common surname in England and southern Wales.
Steve Vander Ark suggested the name might be a reference to characters in The Music Man. The mayor's wife is named Eulalie and her friend is Mrs. Hicks. Typical Rowling trying to find stereotypical names for people from a particular country and going with a very American musical for inspiration.
The exchange, through the magical book, suggests that Hicks and Flamel share a deep understanding of the current events and the stakes involved and that they will have a future role as events unfold. JKR Tweeted “You only see a HINT of Lally in Fantastic Beasts 2. Her true glory is revealed in FB3.”