Would a gravitational wave accelerate a single ball? Suppose I have two balls floating in space. If a gravitational wave with the correct polarization passed by, it would create an oscillating strain causing the balls to accelerate together, then apart, and together, and apart, and so on.
But what if I got rid of one of the two balls? This shouldn't affect the remaining ball's motion, so would it still oscillate back and forth all by itself? And if so, which way would it go; would it choose the prior motion of the left ball or the right ball? Or, maybe the answer is that its motion is undefined because there's nothing to compare its position to, but then couldn't I place a precise accelerometer on the ball to figure out what its acceleration is?
 A: The gravitational wave doesn't accelerate either ball: according to GR each ball follows a geodesic trajectory. However, the distance between those trajectories changes as the wave passes.
Note added in response to comments:
Imagine you're on a perfectly spherical planet. Start walking north at the equator. Follow a geodesic: go straight ahead, accelerating neither left nor right. You'll thus follow a meridian to the north pole.
Have a friend start on the equator on a parallel course. They, too, will follow a meridian to the north pole. So, your paths cross despite the fact that neither accelerated. You apparently accelerate toward each other despite the fact that neither accelerates. No force pulls you together, but you come together anyway.
Note that this happens for any pair of geodesic (great circle) paths on the sphere. They will intersect at two points, but they'll also be parallel at two places. I just chose points that intersect at the pole for easier visualization.
From a geometric point of view, this happens because a sphere is not a flat Euclidean plane. Your Euclidean expectations don't apply.
An accelerometer measures the difference between its motion and free fall. So, if your test masses are free falling, an attached accelerometer will register no acceleration. However, the distance between the free falling masses reflects the geometry. In the GR model of a gravitational wave, that geometry is fluctuating.
