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I'm teaching maths at the 8-16 year level, one-on-one tuition.

What are some good pen-and-paper maths games I can play with these kids?

Googling "math games" just gets me a load of interactive animated junk and most games in this list are not actually educational and most are not really mathematical either (in the high-school-level sense -- they are combinatorial games).

It's important that these games teach or give exercise in something other than arithmetic.

Ideally these games would have an element of competition (teacher and pupil racing to complete something, for example) but I need to be able to invent instances of the game in just a moment's thought.

Any ideas?

We used to do this in my high school math class. I am not sure if it is too much of lower-level math, but we had to take the numbers from the year (i.e. 2, 0, 1, and 9) and then make equations that are equal to the numbers 0 through 2019 (or whatever year it is) using each number only once, but any mathematical functions... It's easy for some numbers, but pretty difficult for others. I realize it is mostly arithmetic but still a cool warm-up math game.

If I’ve understood it well, you want to play some ‘mathematical game’ to introduce/teach a pupil something. I thought of trying to estimate a ‘difficult’ number (as fast as possible) (like $sqrt[3]1.5$) in order to introduce the use of derivatives.

In this example, define $f(x)=sqrt[3]x$.
We know that $f(1)=sqrt[3]1=$1 and that $f’(1)=frac13sqrt[3]x^2$, so a good estimate would be
$$sqrt[3]frac32approx f(1)+frac12cdot f’(1)=1+frac16=1.1666cdots$$

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".

I really like the following two games: We play them a lot just for fun, but not really to teach mathematics.

1) Racetrack, we call it Vectorrace. It's a turn based race game where your change of position is determined by a velocity vector. Each turn you can only change your velocity a little bit, which requires you to think ahead. The educational value lies in explaining the concept of a vector, but also the concept of velocity and acceleration viewed as vectors. Age: $geq$8. (Bonus: Also try the Möbius-strip-variant, where you glue together mutliple sheets of papers to form a Möbius-racetrack. It's awesome to realize that you are now cruising on the other side of the paper.)

2) Higherdimensional Battleship. The pen and paper game battleship is well known. It could teach some young kids the cartesean coordinate grid, but other than that it is not really mathematical. However, when math students are introduced to $n$-dimensional objects with $n>3$, then such a game can help immensely to get a feeling for it. They can realize for example that a 4D-vector is formally just a 4tuple, but there is a sense of adjecency in 4D too. It is a great challenge to come up with a way to represent a 4D grid on paper, (there are many ways to do this,) and when placing the boats at the beginning, the constraint that no boats are allowed to be placed next to each other (even diagonally) is challenging to think about. Age: 8-12 for 2D, >15 for 4D. (In fact many simple pen and paper games can be generalized to higher dimensions. For example: Tic-tac-toe, connect four, ...) (Recommended 4D game size: 5x5x4x4, one 4-ship, one 3-ship, two 2-ships, three 1-ships.)

Have fun!