Eloise is the lady from the shop that Desmond tried to buy the engagement ring from. She denied him, and told him that he was going to the island.
She turned up last epi, doing some intense math to calculate how long Ben had until, well, whatever. I think she is the young Ellie in this episode.
If she is, in fact, the young Ellie (the blonde with the gun on the island), it will be interesting to find out why she's helping Ben after he killed all of the descendants of the circa 1954 Others. I guess, for now, it can be chalked up to their common interest of protecting the island.
It was good to find out more of the happenings on the island.... the Oceanic 6 story-line is kind of blah.