What are gravitational waves made of?

A wave is a traveling distortion. This goes for any type of wave. An ocean wave is a distortion of the water surface. A sound wave is a distortion in air pressure. A light wave is a distortion in electromagnetic fields. A wave is made of the thing that is vibrating--ocean waves are made of water, etc. So, a gravitational wave is made of space and time, since gravity is the effect of space and time warping due to nearby masses.

In General Relativity the effect of gravitation is represented by the stress energy tensor, $T_{μν}$:

$$ R_{μν} - \frac{1}{2}Rg_{μν} + \Lambda g_{μν} = \frac{8\pi G}{c^4}T_{μν} $$

where $R_{μν}$ is the Ricci curvature tensor, $R$ is the scalar curvature, $g_{μν}$ is the metric tensor, $Λ$ is the cosmological constant, $G$ is Newton's gravitational constant, $c$ is the speed of light in vacuum, and $T_{μν}$ is the stress–energy tensor.

Gravitational waves are the sinusoidal changes on space time induced by specific time dependent changes in the stress-energy tensor. In a Newtonian frame it is called acceleration , which is not enough to get a gravitational wave, only asymmetric mass distributions will give gravitational waves.

So it is sinusoidal changes in the stress-energy tensor which distort the space time solutions which fulfill the equation of general relativity.

The waves in general are fluctuations, or ripples in something. Gravitational waves are ripples in space(time).

Now, what is space? Space itself is not defined. What is defined in GR is its certain properties. These properties/attributes include things like tensors, metrics, and manifolds. As space is defined only in terms of these properties, a ripple in space has to be a ripple in value of one or more of these properties. Meaning the value of one or more of these properties is rippling/changing. The ripple in the value is propagating at the speed of light.

So, the gravitational waves are made of fluctuation in the value of certain property of space. I am not certain about role of these terms. But per anna v in her answer, looks like the rippling property is the "stress energy tensor".

In common language, the Stretchiness of space is changing, and that change is propagating at light speed. Because of changed Stretchiness, the behavior of space on its path changes slightly.