The information that the random variable in question "is equal to $X$ with probability $1/2$, and equal to $Y$ with probability $1/2$" is not sufficiently clear, as stated, in order to allow us to provide some definite answer. Proof by example:
1) Assume that the random variable in question, denote it $W$, is defined as:
$$W = \left\{
\begin{array}{lr}
X \;\; \text{with probability } \frac{1}{2}\\
Y \;\;\text{with probability } \frac{1}{2}
\end{array}
\right.\\$$
This is equivalent to define an independent Bernoulli rv $B(p=0.5)$ and set
$$W = X\cdot B + Y\cdot (1-B)$$
It is not difficult to conclude that $W$ will also be a standard normal random variable, with $E(W) = 0$.
2) But define now the rv $Z = \max\{X,Y\}$. Its distribution function is
$F_Z(z) = \Phi(z)^2$, and its density is $f_Z(z) = 2\phi(z)\Phi(z)$, where $\phi(z)$ is the standard normal density, and $\Phi(z)$ the standard normal distribution function. We see that $Z$ is Azzalini's Skew Normal distribution, with location parameter $0$, scale parameter $1$, and shape (or skew or slant) parameter $1$. Then we have $E(Z) = \frac 1{\sqrt {\pi}}$.
Now note that
$$P(X > Y) = \int_{-\infty}^{\infty}\int_{y}^{\infty}\phi(y)\phi(x) dxdy = \int_{-\infty}^{\infty}\phi(y)\cdot [1-\Phi(y)]dy$$
$$=1-E[\Phi(Y)] = 1-\frac 12 = \frac 12$$
by the probability integral transform.
So the event "$Z$ is equal to $X$" has probability equal to $1/2$, and the event "$Z$ is equal to $Y$" has also probability $1/2$.
Therefore, what defines $W$, holds also for $Z$, although they are different random variables, and they are characterized by different distributions... but this means that, at the verbal level, we should clearly distinguish between the statements
$A$: "A random variable is defined as being equal to $X$ with probability $1/2$ and equal to $Y$ with probability $1/2$"
and
$B$: "A random variable has the property of being equal to $X$ with probability $1/2$ and $Y$ with probability $1/2$".
We see that the information, as given in the question, is fatally vague, and could be matched either to statement $A$ or to statement $B$.
If it had been given as statement $A$ above, we could definitely provide an answer. If it had been given as statement $B$ above, we could definitely say that we cannot provide a definite answer.
Essentially, the property in question is not a defining property. To the degree that we are not sure whether the property we want to communicate leads indeed to a unique characterization, we should take the trouble to choose either statement $A$, or statement $B$, and avoid the seemingly transparent statement provided in the question.
