Is the score-function (gradient of likelihood function) a random-variable? 
Wikipedia:The score is the gradient (the vector of partial derivatives) of $\log \mathcal{L}(\theta)$, the natural logarithm of the likelihood function, with respect to an $m$-dimensional parameter vector $\theta$.

The score has a mean (zero) and variance (fisher info).
So, is the score a random variable?

Both the log-likelihood function and the score function can be considered as functions of the parameter vector, based on a fixed observed data vector.  However, we can broaden our interpretation of these functions to consider them to also be functions of the data vector (seeis th related answer).  Taking this broader view of these functions means that the score function can be considered as a random variable by substituting the random data vector into the data $\mathbf{x}$ for the function.  To do this, we take the log-likelihood to be a function of both the data and the parameter:
$$\ell_\mathbf{x}(\theta) = g(\mathbf{x},\theta).$$
The score function is then also a function of both the data and the parameter:
$$s_\mathbf{x}(\theta) = h(\mathbf{x},\theta) = \nabla_\theta g(\mathbf{x},\theta).$$
Using this mapping, we then obtain the random version of the score function by taking $\mathbf{X}$ to be a random vector and using:
$$s_\mathbf{X}(\theta) = h(\mathbf{X},\theta).$$
