During the Regency Period, it was common for women family members to call each other by their christian names (first names). But men, even informally, were more often called by their sir names. Hence, we frequently see even Mr. Bingley refer to his best friend Mr. Darcy as "Darcy." We see an example of this at the Meryton ball. Mr. Bingley says, "Come, Darcy...I must have you dance. I hate to see you standing about by yourself in this stupid manner" (Vol. 1, Ch. 3).
Women who grew up together from childhood, such as sisters or school mates, commonly called each other by their first names ("Correct Forms of Address"). Therefore, in referring to Elizabeth by her first name, and sometimes even to Jane by her first name, rather than by Miss Bennet, Austen is putting the narrator on the same level of familiarity as the rest of her female characters, especially the Bennet sisters. Austen is making the narrator act like an extra sister to the girls or at least as an extra close friend. The effect is that even though Austen uses a third person omniscient narrator that focuses on Elizabeth, because Elizabeth is always addressed informally by the narrator, the reader feels more intimately acquainted with Elizabeth and the other characters.
Also, in not addressing any of the men informally, but rather always as Mr. Bingley or Mr. Darcey, Austen is conforming to the accepted social norms for women. Women only addressed men formally, and as a woman, it makes perfect sense that Austen would have her narrator act as a woman and address the men formally but treat the women as close friends or sisters.