How to generate random points in $\ell_p$ balls?

For arbitrary p, this paper does exactly what you want. Specifically, pick $X_1,\ldots,X_n$ independently with density proportional to $\exp(-|x|^p)$, and $Y$ an independent exponential random variable with mean 1. Then the random vector $$\frac{(X_1,\ldots,X_n)}{(Y+\sum |X_i|^p)^{1/p}}$$ is uniformly distributed in the unit ball of $\ell_p^n$.

The paper also shows how to generate certain other distributions on the $\ell_p^n$ ball by modifying the distribution of $Y$.

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I'll assume that you're looking for a uniformly chosen random point in the ball, since you didn't state otherwise. For p=1, you're asking for a uniform random point in the cross polytope in n dimensions. That is the set

$ C_n = \{ x_1, x_2, \ldots, x_n \in \mathbb{R} : |x_1| + \cdots + |x_n| \le 1 \}. $

By symmetry, it suffices to pick a random point $(X_1, \ldots, X_n)$ from the simplex

$ S_n = \{ x_1, x_2, \ldots, x_n \in \mathbb{R}^+ : x_1 + \cdots + x_n \le 1 \}$

and then flip $n$ independent coins to attach signs to the $x_i$.

From Devroye's book Non-uniform random variable generation (freely available on the web at the link above, see p. 207 near the beginning of Chapter 5), we can pick a point in the simplex uniformly at random by the following procedure:

• let $U_1, \ldots, U_n$ be iid uniform(0,1) random variables
• let $V_1, \ldots, V_n$ be the $U_i$ reordered so that $V_1 \le V_2 \le \cdots \le V_n$ (the "order statistics"); let $V_0 = 0, V_{n+1} = 1$
• let $X_i = V_i - V_{i-1}$

So do this to pick the absolute values of the coordinates of your points; attach signs chosen uniformly at random, and you're done.

This of course relies on the special structure of balls in $\ell^1$; I don't know how to generalize it to arbitrary $p$.