FIR Filter 3dB Frequency
There are several problems with your approach:

The expression $\displaystyle\frac{1+z^{-1}}{2}$ is the transfer function $H(z)$, NOT its magnitude $|H(z)|$.
Consequently, the expression $\displaystyle \frac{1+e^{-j\Omega}}{2}$ is the complex frequency response $H(e^{j\Omega})$, not its magnitude.

So now you have to compute the squared magnitude of $H(e^{j\Omega})$ and set it equal to $\frac12$. This is a basic exercise in complex numbers, so I trust that you can take it from here.
For some magical reason, the right half of you last equation is actually the final correct equation, even though everything before it is wrong. If you think about it for a while, you should figure out that the solution is given by
$$\cos(\Omega_c)=0\tag{1}$$
which certainly does not imply that $\Omega_c=0$. What is it that it does imply?