In a region of uniform electric field of intensity E, an electron of mass $m_e$ is released from rest. The distance travelled by electron in time t is: 

When an electron is discharged from rest into an electric field, it has a mass of m_e and intensity of electric field of E, according to the question. Accordingly, the electron's initial velocity is zero (u=0), thus let e represent its charge.
Write the force this electron felt in the electric field using
$F_{EF}=qE$

$F_{EF}=eE$----------------------------1

The Newton's law, F=ma, is now known to us. This allows us to rewrite equation (1) for the electron in the following way. 

$m_{e}a = eE$-----------------------2

The distance the electron travelled in time t must be determined. We shall use one of the motion equations for this. So let's utilise
$s=ut+\frac{1}{2}a{t}^2$

Let's try replacing these values in the equation.
$s=0\times t+\frac{1}{2}a{t}^2$

Let's make it more simpler.

$s=\frac{1}{2}a{t}^2$

We shall subtract the value of acceleration as follows from this equation.

$a=\frac{2s}{t^2}$---------------------3

Let's now change the value of the variable a from equation (3) to (2). 

$m_e\frac{2s}{t^2}=eE$

Let's remove the distance's' value from the previous phrase.
$s=\frac{eE{t}^2}{2m_e}$
Hence, the correct answer is option B.