Not a compete answer, but observations on the value of $\sigma$ for which the minimum occurs. Throughout I'm assuming the Riemann Hypothesis.
In "Zeros of $\zeta^\prime(s)$ and the Riemann Hypothesis", Spira proves that the Riemann Hypothesis holds if and only if $|\zeta(s)|$ is increasing as $\sigma$ decreases from $1/2$ to $-\infty$ for $t\gg 1$ (He seems to prove $t>165$.) So it suffice to consider the minimum over $1/2\le \sigma\le 1$.
From now on we assume $\zeta(1/2+it)\ne 0$ (else the minimum over $\sigma$ is already $0$ for that $t$.) Then $|\zeta(\sigma+i t)|>0$, and so $\ln|\zeta(\sigma+i t)|$ is defined. Since $\ln$ is increasing and preserves inequalities, the same $\sigma$ minimizes both $|\zeta(\sigma+i t)|$ and $\ln|\zeta(\sigma+i t)|$. There are three possibilities:
I: $\ln|\zeta(\sigma+i t)|$ is increasing on $1/2\le \sigma\le 1$ and the minimum occurs at $\sigma=1/2$.
II: $\ln|\zeta(\sigma+i t)|$ is decreasing on $1/2\le \sigma\le 1$ and the minimum occurs at $\sigma=1$.
III: $\ln|\zeta(\sigma+i t)|$ has a local minimum on $(1/2,1)$. (Conceivably, the global minimum may still occur at an endpoint.) This is the case in which
$$
\frac{d}{d\sigma}\ln|\zeta(s)|=0
\Leftrightarrow \frac{\partial}{\partial \sigma} \text{Re}(\log(\zeta(s)))=0
\Leftrightarrow \text{Re}\left(\frac{\zeta^\prime(s)}{\zeta(s)}\right)=0.
$$
This latter happens when either
IIIa: $\zeta^\prime(s)=0$, or
IIIb: $\arg\left(\frac{\zeta^\prime(s)}{\zeta(s)}\right)=\pm \pi/2.$
The zeros of $\zeta^\prime(s)$ occur discretely (just as do the zeros of $\zeta(s)$). Meanwhile the level curves for $\arg=\pm \pi/2$ will connect the zeros of $\zeta^\prime/\zeta$ (i.e. the zeros of $\zeta^\prime(s)$) with the poles of $\zeta^\prime/\zeta$ (i.e. the zeros of $\zeta(s)$ on the critical line.)
Titchmarsh shows that $\zeta(s)\ne 0$ for $\sigma>3$, and zeros with $\sigma>1$ do exist. But Spira shows in "Zeros of $\zeta^\prime(s)$ in the critical strip" that 'most' zeros are close to $\sigma=1/2$ (in the sense that the number of zeros with $\sigma>1/2+\delta$, $t<T$ is only O(T).) Lots of research has been done on the clustering of the zeros of $\zeta^\prime$ near the critical line.
Edit: For $t\gg1$, possibility I above can not occur. For if it did, then combined with Spira's result above, we would have that $\sigma=1/2$ is a local minimum. But the formula for the logarithmic derivative
$$
\frac{\zeta^\prime(s)}{\zeta(s)}=\log(2\pi)-1-C/2-\frac{1}{s-1}-\frac{\Gamma^\prime(s/2+1)}{2\Gamma(s/2+1)}+\sum_{\rho}\left(\frac{1}{s-\rho}+\frac{1}{\rho}\right),
$$
along with Stirling's formula, gives that
$$
\text{Re}\frac{\zeta^\prime}{\zeta}(1/2+it)\sim -\log(t/2)/2,
$$
and in particular is not $0$.
Case II above should correspond to zeros of $\zeta^\prime(s)$ with $\sigma>1$, or consecutive zeros of $\zeta$ with no zero of $\zeta^\prime$ in between (for otherwise the horizontal line $t=\text{const.}$ should cross a level curve $\arg=\pm \pi/2$.) This latter phenomenon happens at intervals on average of $2\pi/\log(2)\approx 9.064$, due to the different asymptotics of the zeros of $\zeta$ and $\zeta^\prime$.
Final edit: the graphic below shows $t$ on the vertical axis, and, on the horizontal axis, the Hardy function $Z(t)$ (in purple) and the value of $\sigma$ which minimizes $f(t)$. Also marked on the vertical axis are the zeros of $Z(t)$ (in black). The $t$ which correspond to zeros of $\zeta^\prime(s)$ are marked in red when $1/2<\sigma<1$, and in green when $\sigma\ge1$. [Thanks to Ricky Farr for computation of zeros of $\zeta^\prime$.] The data illustrates that minima occuring at $\sigma=1$ tend to be associated to consecutive zeros of $\zeta$ with either no intervening zero of $\zeta^\prime$, or a zero of $\zeta^\prime$ with $\sigma\ge 1$.
(source)
