Let $A = \{1, 2, 3, 4\}$ and $R = \{(1, 2), (2, 3), (1, 4)\}$ be a relation on $A$. We want to find the smallest equivalence relation $S$ on $A$ such that $R \subset S$.

Since $S$ is an equivalence relation, it must be reflexive, symmetric, and transitive.

1. Reflexive: $S$ must contain $\{(1, 1), (2, 2), (3, 3), (4, 4)\}$.

2. $R \subset S$, so $S$ must contain $\{(1, 2), (2, 3), (1, 4)\}$.

3. Symmetric: Since $(1, 2) \in S$, $(2, 1) \in S$. Since $(2, 3) \in S$, $(3, 2) \in S$. Since $(1, 4) \in S$, $(4, 1) \in S$.

4. Transitive: Since $(1, 2) \in S$ and $(2, 3) \in S$, $(1, 3) \in S$. Since $(2, 1) \in S$ and $(1, 4) \in S$, $(2, 4) \in S$. Since $(1, 4) \in S$ and $(4, 1) \in S$, $(1, 1) \in S$.

5. Symmetric (again): Since $(1, 3) \in S$, $(3, 1) \in S$. Since $(2, 4) \in S$, $(4, 2) \in S$.

6. Transitive (again): Since $(2, 3) \in S$ and $(3, 1) \in S$, $(2, 1) \in S$. Since $(1, 2) \in S$ and $(2, 4) \in S$, $(1, 4) \in S$. Since $(2, 1) \in S$ and $(1, 3) \in S$, $(2, 3) \in S$.

So, $S = \{(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 3), (1, 4), (2, 1), (3, 2), (4, 1), (1, 3), (3, 1), (2, 4), (4, 2)\}$.

Now, we need to check if this relation is an equivalence relation.

The equivalence classes are $\{1, 2, 3, 4\}$. We can see that $1, 2, 3$ are related, and $1, 4$ are related. So, $1, 2, 3, 4$ must all be related. Thus, the equivalence classes are $\{1, 2, 3, 4\}$.

Therefore, $S = \{(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4), (4, 1), (4, 2), (4, 3), (4, 4)\}$.

The number of elements in $S$ is $n = 16$.

However, we can also consider the transitive closure of $R$. Since $(1, 2)$ and $(2, 3)$ are in $R$, $(1, 3)$ must be in $S$. Also, $(1, 4)$ is in $R$. So, the elements $1, 2, 3$ are related, and $1, 4$ is related. This means that all elements $1, 2, 3, 4$ are related to each other. Therefore, the equivalence relation $S$ must relate every element to every other element.

Thus, $S = A \times A = \{(a, b) \mid a \in A, b \in A\}$. The number of elements in $S$ is $|A \times A| = |A|^2 = 4^2 = 16$.

Therefore, the minimum value of $n$ is 16.