While this idea is persistently articulated to maintain a particular chronological timeline, the social, physical and scriptural evidence in the Parsha argues otherwise.
First the physical evidence. Rivka performs tasks well outside the range of a three year old. She is sent out by herself to draw water, a task unlikely to be entrusted to a three year old for reasons of physical ability and safety. She then draws water for Eliezer and for all his train of camels, a difficult task for a grown woman, impossible for a three year old girl. When she encounters Yitzchak she dismounts from her camel, possible for a woman, impossible for a girl.
Similarly the social evidence is that Rivka was considered adult enough to be entrusted with adult responsibilities. Her family inquires of her whether she wishes to go, not something they would have bothered with if she was a child. She offers Eliezer the hospitality of her family's house, which she could not have done unless she had enough responsibility in her family's affairs to manage it.
Scripturally Eliezer refers to the girl he is searching for as an Almah when relating it to her family. This excludes a child and the Malbim even states that it especially emphasizes her older age. The description of her doubly emphasizes her virginity, a point that would not have needed to be made if she was three years old. While some point to her nurse accompanying her, it was commonplace for nurses to continue serving their charges for life.
At some point the need to maintain chronology must give way to common sense. Early maturity and various miracles are used to explain away some of the points mentioned above but combined together they become absurd and Rivka was most certainly not three years old.