How to generate new LWE samples
The basic idea is to take random (Gaussian) integer combinations of the given LWE samples, and add a little "smoothing" noise.  This will result in new samples which are statistically close to LWE samples with the same secret, but with a somewhat wider error distribution (by a factor of $\tilde{O}(\sqrt{n})$ for typical parameters). This is essentially Regev's classical BDD-to-LWE reduction on the special family of Ajtai lattices.

More precisely, given LWE samples grouped as $(\mathbf{A} \in \mathbb{Z}_q^{n \times m}, \mathbf{b}^t = \mathbf{s}^t \mathbf{A} + \mathbf{e}^t)$ where $m \approx n \log q$, generate a new sample as $(\mathbf{a}' = \mathbf{A} \cdot \mathbf{r} \in \mathbb{Z}_q^n, b' = \mathbf{b}^t \cdot \mathbf{r} + \tilde{e} \in \mathbb{Z}_q)$, where $\mathbf{r} \in \mathbb{Z}^m$ and $\tilde{e} \in \mathbb{Z}$ have discrete Gaussian distributions of appropriate parameters.  One can show that with high probability over the choice of the original $(\mathbf{A}, \mathbf{b})$, the distribution of $\mathbf{a}'$ is nearly uniform.  Moreover, conditioned on any fixed choice of $\mathbf{a}'$, the distribution of $\mathbf{e}^t \cdot \mathbf{r} + \tilde{e}$ (which is the "error" term in the output sample) is close to a discrete Gaussian.

This procedure is sketched in Gentry-Peikert-Vaikuntanathan, "Trapdoors for Hard Lattices..." (for the tight security proof of the IBE), and is done more formally in Applebaum-Cash-Peikert-Sahai, "Fast Cryptographic Primitives..." (as the encryption algorithm for the KDM-secure public-key scheme).