As discussed in comments, I think for large $n$ the probability that it has a root which is a root of unity is double the probability that 1 is a root. For large $n$, $P(1)$ is a random variable whose distribution is approximately normal and whose variance is $\sigma^2=2n/3$. The probability that $P(1)=0$ is then approximately $1/\sigma\sqrt{2\pi}$. Doubling this gives a probability $\sqrt{3/\pi n}\approx 0.98 n^{-1/2}$. This seems to agree quite well with the final two values in Matt F.'s list:

$$\frac{263}{729}=0.361 \qquad \sqrt{\frac{3}{7\pi}}=0.369$$

$$\frac{2267}{6561}=0.346 \qquad \sqrt{\frac{3}{8\pi}}=0.345$$