Here's what I remember about the game:
In the game you played as two characters who rode on a hand car on a rail way (which means that it was quite literally "on rails"), kinda like this:
You played the game by pumping the handle at a steady pace, and had a little "peddle" at your feet to press in order to either jump out of danger or move to the left or right.
The plot of the game dealt with some time traveler girl trying to get away from some bad guys from her time, and she meets a boy in the past who needs to help her recover a time travel stone and return it to the future. The game had many branching pathways based on whether or not if you manage to recover the stone from the bad guys, or if they manage to steal it away from you. I remember the game used to exhaust me as a kid because of all the arm pumping I had to do, but yet it was so fun I couldn't stop playing. Never got the game's good ending though, the last boss always kicked my butt.
So far I've only seen the game in two locations: my local Chuck-E-Cheese (I have no idea if it's still there, I haven't been there in years nor have I had any reason to go there), and a McDonald's with an arcade down in Orlando. I would really appreciate it if someone could tell me the name of the game, I can't seem to find anything on Google.