The use of the word "response" originates to the application of statistics in medicine: we apply a "treatment" and the patients' bodies "respond to the treatment". In a regression setting the "treatment" would be represented by the regressors - and the dependent variable would measure the "response" to this "treatment".
And in fact what we quantify by the regression technique is the mean response.
We specify a model
$$ y_i = \mathbf x_i'\mathbf \beta + u_i\,,\; i=1,\ldots,n$$
Now take the expected value of the dependent variable conditional on the regressors:
$$ E(y_i\mid\mathbf x_i') = E(\mathbf x_i'\mathbf \beta\mid\mathbf x_i') + E(u_i \mid\mathbf x_i')$$
Given the assumption of strict exogeneity of regressors w.r.t the error term, the last term equals zero. Also, in the second term the regressor vector goes out of the expected value since we condition on it, and $\beta$ is considered a constant (in the classical tradition). So :
$$ E(y_i\mid\mathbf x_i') = \mathbf x_i'\mathbf \beta$$
Since $\beta$ is unknown we estimate it, and we arrive at
$$ \hat y_i = \widehat E(y_i\mid\mathbf x_i') = \mathbf x_i' \hat \beta$$
Even if we knew $\beta$, we would not be able to "calculate"/predict $y$ exactly, because there exists the error term/stochastic-unpredictable disturbance. So, even with known $\beta$ we would be able to obtain just the exact conditional expected value (mean value) of $y$.