Mice may already be "sentient", meaning self-aware and reasonably intelligent.
They show Empathy.
Here is more info on the sentience of mice; it is from PETA, but what they are showing is all proven in a scientific lab setting.
Rats are altruistic, and will help a friend even if it means foregoing a treat. From that link, an excerpt is below.
I'd say mice/rats are already sentient, self-aware, able to imagine a future and act to bring it to pass. What they lack is not sentience but advanced intelligence, the (apparently only human) ability to think in abstractions of abstractions that allow us our science, long term planning, mathematics, etc.
To answer your question: MANY animals are already sentient; and this does not depend upon their brain size at all. Our recursive abstractive intelligence, however, has apparently evolved only once in the approximately one billion years of life on Earth. I imagine any sentient creature (dogs, porpoises, mice, elephants, apes) has the potential to develop recursive abstractive intelligence, but it might be also be a once-in-ten-billion-years event.
In the new study, Mason, Bartal and University of Chicago colleague Jean Decety placed pairs of rats in Plexiglass pens. One rat was trapped in a cage in the middle of the pen, whereas the other rat was free to run around. Most free rats circled their imprisoned peer, gnawing at the cage and sticking their paws, noses and whiskers through any openings. After a week of trial and error, 23 of the 30 rats in the experiment learned to open the cage and free their peers by head-butting the cage door or leaning their full weight against the door until it tipped over. (The door could only be opened from the outside.) At first the rats were startled by the noise of the toppling door. Eventually, however, they stopped showing surprise, which suggests that they fully intended to push the door aside. Further, the rodents showed no interest in opening empty cages or in those containing toy rats, indicating that a break out was their genuine goal.
In this first set of experiments, most rats seemed quite willing to help their peers, but Mason wanted to give them a tougher test. She placed rats in a Plexiglass pen with two cages: in one was another rat, in the other was a pile of five milk chocolate chips—a favorite snack of these particular rodents. The unrestricted rats could easily have eaten the chocolate themselves before freeing their peers or been so distracted by the sweets that they would neglect their imprisoned friends. Instead, most of the rats opened both cages and shared in the chocolate chip feast.
"In our lab we called it the 'chocolate versus pal' experiment," Mason says. "The rat could have put his butt in the opening of the cage containing chocolate to block the other guy, but he didn't. They were sharing food with their pals. In rat land, that is big—I was shocked." Mason says that free rats typically took the chocolate out of the cage before eating it and that sometimes the free rats placed the chocolate chips in front of or very near their recently sprung peers, "as if delivering it."