For example, some textbooks will say:
If the null hypothesis claims that there's no effect/change, how could the null hypothesis be anything other than $H_o=μ$?
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I think the issue is over acceptance of a rule of thumb. That's the problem with thumbs making rules.
My guess is that you've heard something like "the null hypothesis is that there's no effect". But it would be much more accurate to say something along the lines of "often, the null hypothesis is that there's no effect". If $\mu$ is the effect of interest, then certainly $H_o: \mu = 0$ can be correctly interpreted as the null hypothesis being that there's no effect (to be pedantic and circular, this is assuming $\mu=0$ implies no effect. This would not be the case if $\mu$ was a multiplicative factor, for example).
But hypothesis testing is much more general than that special case. You are really testing one set of potential parameters, whether it be $\mu = 0$ or $\mu \leq 0$ against another set of parameters. In theory, there's nothing that limits the structure of the two sets of hypothesis that you choice to construct.
That's theory. Let's talk practice. Recall the fact that we never accept the null hypothesis, but rather at best fail to reject it. On the other hand, through hypothesis testing, we can observe enough evidence to conclude that the parameter belongs to the alternative hypothesis set. Because of this, it makes sense to set up your hypotheses such that from a scientific standpoint, it is very interesting to know that the parameter is in the alternative hypothesis set. So when you design your hypothesis test, your alternative hypothesis should be what your interested in showing, i.e. your drug has a positive effect. As such, your null hypothesis should be the "uninteresting" results, i.e. your drug does not have a positive result.
The null hypothesis can be anything at all. It has nothing to do with "nullness" of effect. Usual tests are built to have an arbitrarily low type I error (ie. $\alpha$), so that we don't mistake the null hypothesis being false when in fact it is true.
For example, in a correlations test we set $H_0: \rho = 0$, so that:
In conclusion: the null hypothesis should be the one you care about the most not rejecting when it is true.
You talk about one and two tailed tests ,the one-tailed tests allow for the possibility of an effect in just one direction , where with two-tailed test you are testing for the possibility of an effect in two directions , so when you reject the null hypothesis in one tailed test you reject that there is no effect and you reject that there is an effect in the direction which you do not care about it.So in your case $$H_o:\mu≤0$$ when you reject $H0$ you reject both "There is no effect " and "There is a negative effect ".Here there is a nice discussion about One-Tailed vs Two-Tailed Tests? and how a two-tailed test splits your significance level .