How does the overdrive voltage relate to the transistor size?
The overdrive voltage is defined from the gate-source voltage and threshold voltage as:

$$ V_{ov} = V_{gs} - V_{th} $$

If you assume \$V_{th}\$ is constant with respect to sizing (a normal first order approximation) then the answer to your question is: The overdrive voltage is not related to the transistor size.

Except, if the device you are varying is already connected in a circuit, allowing \$V_{gs}\$ to change depending on the transistor size. 

If the device is connected so \$V_{gs}\$ may vary, but \$I_{ds}\$ is fixed then you can reason :


If the width increases, \$V_{ov}\$ will go down.
If the length increases, \$V_{ov}\$ will go up.


If \$I_{ds}\$ is fixed and the device is operating in saturation then the following approximation might be useful : $$V_{ov}\propto\sqrt{\frac{L}{W}}$$

If \$I_{ds}\$ is fixed and the device is operating in subthreshold then the following approximation might be useful : $$V_{ov}\propto\log{\frac{L}{W}}$$

These approximations don't take into account variations that occur due to changed \$V_{ds}\$ and \$V_{sb}\$, or changes in \$V_{th}\$. To get a more detailed result take your favorite MOSFET equation and extract \$V_{ov}\$ from it.