OK, it looks like I, indeed, need to spell a few things out.

First, continuous functions with Hausdorff dimension of the graph greater than $1$ exist. I'll skip this part.

Let $g$ be any such function. It can be written as the sum of a uniformly convergent series $\sum_k g_k$ of Lipschitz functions such that $\|g_k\|_\infty\le 2^{-k}$. If it is not already constructed like that, then just consider any sequence of piecewise linear approximations $G_k$ such that $\|G_k-g\|\le 2^{-k-2}$ and put $g_1=G_1$, $g_k=G_k-G_{k-1}$ for $k\ge 2$. Of course, the Lipschitz constants $L_k$ of $g_k$ will grow pretty fast.

Now let the first block be just $\{1\}$. Suppose that we have already constructed the blocks $J_1,\dots,J_n$ and $N$ is the last index of $J_n$. Let $L=\sum_{k=1}^N L_k$. Then the sum of $g_k$ over any subset of the union of the first $n$ blocks (which is just $[1,N]$) is $L$-Lipschitz, so for every $\delta>0$, we can cover the graph of that sum by about $L\delta^{-1}$ disks of radius $\delta$. Choose $\delta_n\in(0,2^{-n})$ so small that $L\delta_n^{-1}\le \delta_n^{-p_n}$ where $p_n\in(1,2)$ is some fixed sequence tending to $1$. Now let the $n+1$-st block to be $J_{n+1}=[N,M]$ where $M$ satisfies $2^{-M}\le\delta_n$. Then the graph of the sum of $g_k$ over any subset of indices disjoint with $J_{n+1}$ can be covered by $\delta_n^{-p_n}$ balls of radius $3\delta_n$, say (the tail beyond $M$ is just too small to really matter on that scale).

Now, by construction, the sum of $g_k$ over odd blocks $J_n$ has a graph that can be covered by $\delta_n^{-p_n}$ balls of radius $3\delta_n$ for all odd $n$. Since $\delta_n\to 0$ and $p_n\to 1$, we conclude that its Hausdorff (or even lower box) dimension is $1$. The same is true for the sum over even blocks.

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