Is the following construction of a secure PRG is also a secure PRG? Given a secure PRG $G :\text{{0,1}}^n \rightarrow \text{{0,1}}^{2n}$, I've constructed the following PRG: 
$$
G'(x) = 
\begin{cases}
0^{2n} &amp; \text{if x is palindrome;}\\
G(x) &amp; \text{otherwise;}\\
\end{cases}
$$
My intuition is that $G'$ is secure because it's "predictable" only for negligible portion of the inputs, so I tried to write a formal reduction. 
I tried showing that given an efficient distinguisher $D'$ for $G'$, I can construct an efficient distinguisher $D$ for $G$. The immediate issue I faced is that $D$ receives as input either a random string $r$ of length $2n$, or $G(s)$ for a random $s$ of length $n$, and In the later case I'm having trouble rearranging $G(s)$ to be "suitable" for $D'$. (which I'm using as a black box) 
Is my intuition about $G'$ security correct? If so, what's the way of proving its' security?
Indeed, your intuition is perfectly correct and the distinguisher $D$ you're trying to build is basically $D'$ itself.
To see that this is the case, simply divide the experiment for $G'$ into two cases: either the chosen input is palindrome or it is not.
The first case occurs with negligible probability, so we can focus in the second case, but now it's easy since this case corresponds to the original experiment for $G$.

If you want to be formal, just keep in mind that
\begin{multline*}
\mathbb{P}\left[D'(G'(x)) = 1\right] = \mathbb{P}\left[D'(G'(x)) = 1|x\text{ is palindrome}\right]\cdot\overbrace{\mathbb{P}\left[x\text{ is palindrome}\right]}^{\delta}
\\ + \mathbb{P}\left[D'(G'(x)) = 1|x\text{ is not palindrome}\right]\cdot\underbrace{\mathbb{P}\left[x\text{ is not palindrome}\right]}_{1-\delta}
\end{multline*}
and 
$$\mathbb{P}\left[D'(r) = 1\right] = 
\delta\cdot\mathbb{P}\left[D'(r) = 1\right] + (1-\delta)\cdot\mathbb{P}\left[D'(r) = 1\right].$$
Now, when you take the difference
$$\begin{align*}
\left|\mathbb{P}\left[D'(G'(x)) = 1\right] - \mathbb{P}\left[D'(r) = 1\right]\right|
\end{align*}$$
try to associate terms appropriately and apply some straightforward inequalities to show this is negligible.