Given
$$X \sim Bin(n, 0.5)\ \ \text{with PDF}\ \ P(X=x)={n \choose x}\left(\frac{1}{2}\right)^{x}\left(\frac{1}{2}\right)^{n-x} = {n \choose x}\frac{1}{2^{n}},$$
how do I find the probability mass function $p(y) = P(Y = y)$ given that $2Y = X$?
$$2Y = X \iff Y = \frac{X}{2}$$
First note that what you are looking for is NOT a density function but a Probability Mass Function.
If you are looking for the pmf of $Y=X/2$, it is always a binomial but with the modified support, thus
$$Y=\{0;0.5;1;1.5;\dots ;n/2\}$$
with the same probabilities as the starting Binomial:
$$\left\{\frac{1}{2^n};\frac{n}{2^n};\frac{n(n-1)}{2^{n+1}};\frac{n(n-1)(n-2)}{3\cdot2^{n+1}};\dots;\frac{1}{2^n} \right\}$$
respectively