When $n$ is divided by $14$, the remainder is $10$. What is the remainder when $n$ is divided by $7$? I need to explain this to someone who hasn't taken a math course for 5 years. She is good with her algebra.
This was my attempt:

Here's how this question works. To motivate what I'll be doing,
  consider  \begin{equation*} \dfrac{5}{3} = 1 + \dfrac{2}{3}\text{.}
\end{equation*} This is because when 5 is divided by 3, 3 goes into 5
  once (hence the $1$ term) and there is a remainder of $2$ (hence the
  $\dfrac{2}{3}$ term). Note the following: every division problem can
  be decomposed into an integer (the $1$ in this case) plus a fraction,
  with the denominator being what you divide by (the $3$ in this
  case).
So, when $n$ is divided by 14, the remainder is 10. This can be
  written as  \begin{equation*} \dfrac{n}{14} = a + \dfrac{10}{14}
\end{equation*} where $a$ is an integer. 
We want to find the remainder when $n$ is divided by 7, which I'll
  call $r$. So \begin{equation*} \dfrac{n}{7} = b + \dfrac{r}{7}\text{,}
\end{equation*}  where $b$ is an integer. 
Here's the key point to notice: notice that  \begin{equation*}
\dfrac{n}{7} = \dfrac{2n}{14} = 2\left(\dfrac{n}{14}\right)\text{.}
\end{equation*} This is because $\dfrac{1}{7} = \dfrac{2}{14}$. 
Thus, \begin{equation*} \dfrac{n}{7} = 2\left(\dfrac{n}{14}\right) =
2\left(a + \dfrac{10}{14}\right) = 2a + 2\left(\dfrac{10}{14}\right) =
2a + \dfrac{10}{7} = 2a + \dfrac{7}{7} + \dfrac{3}{7} = (2a+1) +
\dfrac{3}{7}\text{.} \end{equation*} So, since $a$ is an integer, $2a
+ 1$ is an integer, which is our $b$ from the original equation. Thus, $r = 3$.

To her, this method was not very intuitive. She did understand the explanation. Are there any suggestions for how I can explain this in another way?

We have $n=14a+10$ for some integer $a$. If we take this equation modulo $7$ we have that $n \equiv 10 \mod 7 \equiv 3 \mod 7$ (This because $14 \equiv 0 \mod 7$ and $10 \equiv 3 \mod 7$). You should look up modular arithmetic
where you will see how to deal with this sort of questions. Modular arithmetic is very important in much branches of mathematics mostly in structures that have a finite number of elements in them (finite groups, rings, fields,...).
