3. Among all lines passing through the point $(\sqrt{1994}, 0)$, the number of lines that pass through two different rational points is
3. 1 line.

Obviously, the line $l$ that meets the conditions is not parallel to the $y$-axis, so we can set its equation as $y=k(x-\sqrt{1994})$. Since $l$ passes through two rational points, let's assume they are $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)$. If $k$ is a rational number, $y_{1}=k\left(x_{1}-\sqrt{1994}\right)$ is a rational number, while $x_{1}-\sqrt{1994}$ is an irrational number, so it must be that $k=0$. If $k$ is an irrational number, it contradicts the fact that $\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$ is a rational number. Therefore, the only possibility is $k=0$, and there is only 1 line that satisfies the condition.