How to know the number of constants of a free particle?

Landau and Lifshitz distinguish between constants of motion and conserved quantities--the former being mere mathematical results of integration and the latter being profound statements regarding unchanging quantities in physical systems (energy, momentum, etc.). The $2s-1$ quantity is a count of the former.

For a free particle, its motion can be given by $vec{x}(t) = vec{x}_0 + vec{v}t$, where $vec{x}_0$ is the position when time $t=0$ and $vec{v}$ is the velocity. There are six constants of motion: three in $vec{x}_0$ and three in $vec{v}$. However, this is, in a sense, over-specified since time does not explicitly occur in the equations of motion (i.e., the original Lagrangian). Choosing another initial position, $vec{x}(t) = vec{x}_0' + vec{v}t$ results in the exact same motion as long as $vec{x}_0' = vec{x}_0 + vec{v}k$ for some value of $k$. The space of constants that don't make a difference is parameterized by a one-dimensional scalar $k$, so the space of constants of motion that do make a difference and are independent is reduced by one.

Since a translation in time is an unimportant transformation (time is homogeneous, so the choice of what $t=0$ means is arbitrary), we can create a constant of motion associated with the choice of the initial time coordinate $t_0$, leaving $2s-1$ constants that depend only on the position and velocity coordinates ($q_i$ and $dot{q}_i$ for $i in 1, 2, ... , s$ in the textbook notation). The latter set of constants contain all of the interesting physics.

Here's an example of a set of five ($2s-1$ where $s = 3$) constants of motion that completely define a free particle:

1. Velocity along the x-axis,
2. Velocity along the y-axis,
3. Velocity along the z-axis,
4. The distance of closest approach to the origin,
5. The angle of the position vector from the positive z-axis at the closest approach to the origin.

These five numbers define the motion of the particle unambiguously in a timeless way.

It's true there are 7 conserved quantities. But for a free particle begin{equation*} E=frac{1}{2}mdot{mathbf{q}}^{2}=frac{mathbf{p}^{2}}{2m} end{equation*} so $E$ can be written in terms of $mathbf{p}$. So we have now 6 conserved quantities. The other relation you need is $mathbf{p}cdotmathbf{L}=0$ where $mathbf{p}$ is the linear momentum and $mathbf{L}$ is the angular momentum, which implies that one component of them can be expressed in terms of the others. So the independent conserved quantities are back to 5.