Let $s(n)$ denote the sum of the digits of $n$. If $3$ divides $n$, then $3$ divides $s(n)$. The minimum possible value of $s(n)$ that we can construct is obviously $3$. It is easy to verify that $39$ divides $10^2 + 10^4 + 10^6$. Therefore, the minimum value of $s(n)$ is $3$.

The answer is $\boxed 3$. Clearly $n$ must have a sum of digits of at least $3$ since it is a multiple of $3$ ($3 \mid 39$). Example constructions are \[10^{x+4} + 10^{x+2} + 10^x = 10^x(10^4 + 10^2 + 1) = 10^x \cdot 39 \cdot 259\]for any nonnegative integer $x$.