Adding AWGN noise with a correct noise power to the signal
The kind of AWGN channel you're simulating is sometimes called a "discrete time" channel. That means that you're going to simulate the bit error rate at the output of a decision device, the input to which is produced by a correlator. This is more easily explained by looking into the transmission and reception of a single bit. I'll assume BPSK.

Let's say that you transmit the physical signal $a_0p(t)$, where $a_0$ is either $A$ or $-A$, and $p(t)$ is a pulse of unit energy. To make things easier we'll assume $p(t)$ is symmetrical, that is, $p(t)=p(-t)$ (this is true for all practical systems I know). Note that, in this scenario, the bit energy is $E_b=A^2$.

Assuming that there is no noise, the received signal is $r(t)=a_0p(t)$. The receiver correlates $r(t)$ with the transmitted pulse $p(t)$. The output of the correlator is $$\int_{-\infty}^\infty a_0 p(t)p(t)\,dt=a_0,$$ where I've used the fact that the pulse has unit energy.

Now assume that the input to the correlator is pure Gaussian white noise $n(t)$ with PSD $N_0/2$ and zero mean. Then, the correlator's output is $$n=\int_{-\infty}^\infty n(t)p(t).$$ By the Wiener-Khinchine theorem, $n$ is a zero-mean Gaussian random variable with variance $\sigma_n^2=N_0/2$.

In the case of having signal plus noise at the receiver's input, the output will be $$r=a_0+n$$ because the correlator is linear. Using the value of $r$, a decision can be made on whether $A$ or $-A$ was transmitted.

We define the SNR at the correlator's output as the bit energy divided by the noise variance: $$\text{SNR}=\frac{a_0^2}{\sigma_n^2}=\frac{2E_b}{N_0}=\frac{2A^2}{N_0}.$$

Notes:


When you plot the BER, you'll likely use $E_b/N_0$ in the horizontal axis. The reason is that in quadrature modulations, the total noise power is $N_0$, not $N_0/2$.
Practical systems use a matched filter instead of a correlator. The results are exactly the same.
In a practical system there will be delay, you transmit multiple pulses instead of only one, etc. Those details are not essential to understanding how the SNR is measured in your simulation.