= Solution
Let $S$ be the support of the weight-$d$ codeword $x$, and let $\pi_S$ delete all coordinates in $S$. Its kernel consists of codewords supported in $S$. If $0\ne y\in\ker\pi_S$, then $\operatorname{wt}(y)\geq d$ but $|S|=d$, so $y$ is nonzero in every coordinate of $S$. Over $\mathbb F_2$ this forces $y=x$. Hence
$$
\ker\pi_S=\langle x\rangle
$$
has dimension one, and the <punctured code> $C'=\pi_S(C)$ has length $n-d$ and rank $k-1$ by the <rank-nullity theorem>.
To bound its minimum distance, take $y\in C$ whose puncture $y'$ is nonzero. If $y$ has $r$ nonzero coordinates in $S$, then $y+x$ has $d-r$ there, while both words have the same $\operatorname{wt}(y')$ nonzero coordinates outside $S$. Both $y$ and $y+x$ are nonzero codewords, so
$$
\operatorname{wt}(y')+r\geq d,
\qquad
\operatorname{wt}(y')+d-r\geq d.
$$
Adding gives $2\operatorname{wt}(y')\geq d$, and therefore
$$
\boxed{d'\geq\left\lceil\frac d2\right\rceil}.
$$
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