It depends on where on Mars you toss the coin, and how high you toss it.
In a rotating frame of reference, an object in motion appears to be affected by a pair of fictitious forces - the centrifugal force, and the Coriolis force. Their magnitude is given by
$$\mathbf{\vec{F_{centrifugal}}}=m\mathbf{\vec\omega\times(\vec\omega\times\vec{r})}\\ \mathbf{\vec{F_{Coriolis}}}=-2m\mathbf{\vec\Omega}\times\mathbf{\vec{v}}$$
The question is - when are these forces sufficient to move the coin "away from your hand" - in other words, for what initial velocity $v$ is the total displacement of the coin greater than 10 cm (as a rough estimate of what "back in your hand" might look like; obviously you can change the numbers).
The centrifugal force is only observed when the particle is rotating at the velocity of the frame of reference - once the particle is in free fall, it no longer moves along with the rotating frame of reference and the centrifugal force "disappears". The Coriolis force is strongest at the equator, becoming zero at the pole; it is a function of the velocity of the coin. We will calculate the expression as a function of latitude - recognizing that it will be a maximum at the equator.
As a simplifying assumption, we assume the change in height is sufficiently small that we ignore changes in the force of gravity; we also ignore all atmospheric drag (in particular, the wind; if the opening scene of "The Martian" were to be believed, it can get pretty windy on the Red Planet.) Finally we will assume that any horizontal velocity will be small - we ignore it for calculating the Coriolis force, but integrate it to obtain the displacement.
The vertical velocity is given by
$$v = v_0 - g\cdot t$$
and the total time taken is $t_t=\frac{2v_0}{g}$. At any moment, the Coriolis acceleration is
$$a_C=2\mathbf{\Omega}~v\cos\theta$$
Integrating once, we get
$$v_h = \int a\cdot dt \\ = 2\mathbf{\Omega}\cos\theta\int_0^t(v_0-gt)dt\\ = 2\mathbf{\Omega}\cos\theta\left(v_0 t-\frac12 gt^2\right)$$
And for the displacement
$$x_h = \int v_h dt \\ = 2\mathbf{\Omega}\cos\theta\int_0^t \left(v_0 t-\frac12 gt^2\right)dt\\ = 2\mathbf{\Omega}\cos\theta \left(\frac12 v_0 t^2-\frac16 gt^3\right)$$
Substituting for $t = \frac{2v_0}{g}$ we get
$$x_h = 2\mathbf{\Omega}\cos\theta v_0^3\left(\frac{4}{g^2} - \frac{4}{3 g^2}\right)\\ = \frac{16\mathbf{\Omega}\cos\theta v_0^3}{3g^2}$$
The sidereal day of Mars is 24 hours, 37 minutes and 22 seconds - so $\Omega = 7.088\cdot 10^{-5}/s$ and the acceleration of gravity $g = 3.71 m/s^2$. Plugging these values into the above equation, we find $x_h = 2.75\cdot 10^{-5}v_0^3 m$, where velocity is in m/s. From this it follows that you would have to toss the coin with an initial velocity of about 15 m/s for the Coriolis effect to be sufficient to deflect the coin by 10 cm before it comes back down.
On Earth, such a toss would result in a coin that flies for about 3 seconds, reaching a height of about 11 m. It is conceivable that someone could toss a coin that high - but I've never seen it.
AFTERTHOUGHT
Your definition of "vertical" needs to be carefully thought through. There is a North-South component of the centrifugal "force" that is strongest at 45° latitude, and that will cause a mass on a string to hang in a direction that is not-quite-vertical. If you launch your coin in that direction, you will not observe a significant North-South deflection during flight, but if you were to toss the coin "vertically" (in a straight line away from the center of Mars), there will in fact be a small deviation. The relative magnitude of the centrifugal force and gravity can be computed from
$$\begin{align}a_c &= \mathbf{\Omega^2}R\sin\theta\cos\theta \\ &= \frac12 \mathbf{\Omega^2}R\\ &= 8.5~\rm{mm/s^2}\end{align}$$
If you toss the coin at 15 m/s, it will be in the air for approximately 8 seconds. In that time, the above acceleration will give rise to a displacement of about 27 cm. This shows that your definition of "vertical" really does matter (depending on the latitude - it doesn't matter at the poles or the equator, but it is significant at the intermediate latitudes, reaching a maximum at 45° latitude).