2. -1 or 3.

From $S^{2}=\frac{1}{6}\left(x_{1}^{2}+x_{2}^{2}+\cdots+x_{6}^{2}-6 \bar{x}^{2}\right)$, we know $6 \bar{x}^{2}=24, \bar{x}= \pm 2$.
Therefore, the required average of the data is $\bar{x}+1=-1$ or 3.
2. Given a set of data $x_{1}, x_{2}, \cdots, x_{6}$ with the variance
$$
S^{2}=\frac{1}{6}\left(x_{1}^{2}+x_{2}^{2}+\cdots+x_{6}^{2}-24\right) \text {. }
$$

then the mean of the data $x_{1}+1, x_{2}+1, \cdots, x_{6}+1$ is $\qquad$