Draw DFA that accepts concatenated binary string with condition
Hint: $3\operatorname{value}(u)=\operatorname{value}(u)+\operatorname{value}(0u)$. Then you're just doing a simple addition with carry.



By the way, there's a much stronger result: For any first order formula $\varphi(x_1,\dots,x_n)$ of the Presburger arithmetic, there is an automaton $A$ over $\{0,1\}^n$ so that $A$ accepts $u$ iff $\varphi(\operatorname{value}_1(u),\dots,\operatorname{value}_n(u))$ is true (where $\operatorname{value}_i(u)$ is the same value that you defined except that since each "letter" of $\{0,1\}^n$ contains several digits, it only looks at the $i^{th}$).

Your question is a special case of this where the formula is $\varphi(a,b)= ``a=3b"$. (Kind of: if $L$ is what you get for this formula, then you'd need to take $f(L)$ where $f$ is the morphism defined by $(b_1,b_2)\mapsto b_1b_2$).