Ben,
$$\|x\|^\prime = \inf\Big\{ \big(\sum_{i=1}^n \|x\|^p\big)^{1/p}\colon \sum_{i=1}^n x_i = x, x_i\in X, n\in \mathbb N \Big\}\qquad (x\in X)$$$$\|x\|^\prime = \inf\Big\{ \big(\sum_{i=1}^n \|x_i\|^p\big)^{1/p}\colon \sum_{i=1}^n x_i = x, x_i\in X, n\in \mathbb N \Big\}\qquad (x\in X)$$
is the standard $p$-convex renorming. The hardish part is to find a suitable $p$. You will find more details in Kalton & Peck's An F-space sampler.