I find this question both interesting and impossible to answer - impossible because you don't say anything about the goals of your one-on-one tuition. Is this to help weak or failing students in their regular classes? To point toward doing well on some standardized test? To show what mathematics is (not just arithmetic and algorithms) and how much fun it can be? Moreover, ages 8-16 is a very large range. Games appropriate at one end of the range won't work at the other.
I think the best games are actually investigations, with exploratory examples followed by generalizations and (semi)formal arguments at an appropriate level. For example:
If there are $n$ people in a room and everyone shakes hands with everyone else how many handshakes will there be? (Leading to the sum $1+2+\cdots+n$. Easy to see the recursion in a table of small values, harder to find the answer for $n=100$.)
If you are allowed one of each kind of integer weight and want the smallest set of weights to weigh any integer value up to $n$ what should the weights be? (Powers of $2$, base $2$ representation.)
What's the answer to the previous question if you are allowed to put weights on both sides of the balance?
Explore the combinatorics of the face structure of the cube in $n$ dimensions, starting with counts of faces of various dimensions for the known dimensions $n = 0,1,2,3$. Build models of the three dimensional views of the tesseract.
Lots more along these lines ... searching for "recreational mathematics" will probably find better resources than your search for "mathematical games".