Answer: C

Small Hint:

A sum of $1994$ is possible only when $6n\ge1994$

Big Hint:

Replacing every die value $d$ by $7-d$ pairs sums $T$ and $7n-T$

Solution:

The smallest feasible number of dice is $$n=\left\lceil\frac{1994}{6}\right\rceil=333.$$ Replacing every die result $d$ by $7-d$ is a bijection between outcomes of sum $T$ and outcomes of sum $7n-T.$ Thus for $n=333,$ the sum paired with $1994$ is $$S=7(333)-1994=337.$$ Dice-sum counts increase strictly from the minimum sum up to the mean, so for any feasible $n$ no sum below $337$ can match the count at $1994.$ Thus the smallest value is $337,$ and the correct answer is C.