There is no lower bound for Re$(\zeta(1+it))$. This can be found in Lamzouri's paper http://www.math.uiuc.edu/~lamzouri/distribzeta.pdf (see e.g. page 3, formula (5)). (Choose $\tau$ large enough, $\theta_1=3\pi/4$ and $\theta_2=5\pi/4$ for and substract). The results of Lamzhouri Lamzouri however also implies that on average the argument of $\zeta(1+it)$ is small and that Re$(\zeta(1+it))$ is positive more often than it is negative.
There is no lower bound for Re$(\zeta(1+it))$. This can be found in Lamzouri's paper http://www.math.uiuc.edu/~lamzouri/distribzeta.pdf (see e.g. page 3, formula (5)). (Choose $\tau$ large enough, $\theta_1=3\pi/4$ and $\theta_2=5\pi/4$ for and substract). The results of Lamzhouri however also implies that on average the argument of $\zeta(1+it)$ is small and that Re$(\zeta(1+it))$ is positive more often than it is negative.