The others all have good advice, but I'd like to add this: remember to act.
While there is a story being told by the playwright, it is important to remember in any size role that the story YOU are telling is YOUR CHARACTER's story. You should be acting just as much in a smaller role as you would in a larger one.
When you're on stage, be in the moment and react to the action honestly as your character would. Even if you don't have a line of dialogue, you're still on stage for a reason and you still have thoughts and feelings about the things going on around you, and you should concentrate on playing them in a believable way (without upstaging others or pulling focus when you shouldn't).
That means doing homework, though, even if it's just time spent thinking about the role. Who the heck IS Cousin Mary? What's going on in her life? Since it sounds like you don't get much help from the script in answering that question, you get to make up the answers yourself. And there may be more clues in the original Mark Twain novels or in Twain's personal history, so doing some reading and research couldn't hurt.
When you're on stage, why are you there? Where did you just come from and where are you going next? What is your relationship to every other character in the show? What is it that you want most (in that scene and in the whole show)? What are you most afraid of? Are you nice? Are you mean? Are you hiding something from one or more other characters? Are you in love with another character? What is your own character arc? Who are you at the beginning of the play and who are you at the end? Did you change/learn/grow/win/lose? How and why?
Yes, it's a small role. And yes, it can feel like a drag if you've played larger roles. What it means, though, is that you have to be better at managing yourself--you're not going to get much character help from the script and you're not going to get much attention from the director during rehearsals. It is, however, a very good way to prove to yourself and others that you're actually an actor, and not just someone who stands on a stage and says words.
I'd also suggest you stop counting your lines (although, of course, sometimes there are so few you notice, anyway). It doesn't help anything, and can only make things worse for yourself or between you and other actors. Why bother?
So the play is called "Tom Sawyer" and there are a half-dozen actors in a cast of forty that speak the bulk of the dialogue. So what? None of that means that Cousin Mary can't be the most interesting character in the show. It's your job to make that happen. Doing it well will make the whole show better.
Break a leg.