Paraphrasing your question: How can a lack of cortisol apparently have the same effect as a surplus thereof?
The book you refer to is a textbook listing symptoms of Addison's disease, without any explanation of that specific entry in the list of symptoms: "weakening of the immune system".
However, as there is no further explaining of "weakening" that expression might be used as a synonym for "lack of strength of the immune response", in other words, saying: "Lack of cortisol weakens the immune system" means, in the context of your quote, a lack of success in fighting inflammation. It would not refer to a weakening of the number of immune cells that is said to be effect of more, not less cortisol. That would be some coherent understanding: Intake of cortisol fights inflammation, hypo-levels of cortisol "weakens" those forces that either fight or inhibit inflammation, "weakens the immune system" in that sense.
In my opinion it is legitimate to claim some error or ambiguity on the side of the publishers, and your question is valid.
Cp. Wikipedia, Addison's disease, which does not list any effect on the immune response as a symptom of that disease.
Conversely, in the case of too much, not to less cortisol, Cushing syndrome, in literature you may easily find some wording like "impaired immune regulation, and defective immune response" with high, not low, dosis of cortisol (Cushing's disease, (e.g.) Hasenmajer et al.. To parallel your question: How some enhancement of immune response against inflammation (that's what cortisol is for) can possibly be described as "impairment" or "defective".
The cause for Addison's disease is seen in a "weak" immune system that attacks cells secreting cortisol, and that might be a source for erroneously exchanging cause and effect (causes for Addison's to differentiate from effects Addison's is being cause of), and the following might be some other illustration:
"...JFK had in fact been diagnosed with Addison's Disease. (...) predicted in 1948 that JFK had only a year to live), since it involved an impairment of the adrenal glands, and a weakening of the immune system.", from Sparknotes on J.F. Kennedy.
My internet research brought up no clear sources claiming any dampening or weakening of the immune response caused by a lack of cortisol due to Addison's disease.
Thus, answering your question, there should be none at all, no "mechanism of hyposecretion of cortisol leading to a depressed immune system". You rightfully asked.
For a corroboration of this answer see, e.g.,
Patterson et al., Cortisol Patterns Are Associated with T Cell Activation in HIV, PLOS One 2013
"We found that lower morning cortisol and flatter diurnal rhythms are associated with greater activation of both CD4+ and CD8+ T cell subsets (...). To our knowledge, this is the first report of the relationship between diurnal cortisol patterns and immune activation in HIV+ subjects, and the results point to the HPA axis as an important regulator of immune activation in persons with HIV."
This speaks in favour of low cortisol levels (as with Addison's, your textbook reference) not being associated with a "weakening", but with an "activation" of lymphocyte, i.e. of t-cells (which then might become auto-aggressive, thus, the result may be termed "weakening of the system" in the sense of dysregulating it). That is consistent with assuming low cortisol (as with Addison's) leading to autoimmunity and inflammation which then might be counter-balanced by high cortisol that is known to "weaken" t-cells that have been activated (not weakened) by low cortison (as with Addison's disease). This should confirm the answer given.
"Weakening" in the text you refer to must be, considering the foregoing, be understood as "over-activating", which makes "depressed" appear as incoherent, as just the opposite.
That statement, in my interpretation, might be some quote from the past, the 50ies or the times of JFK, when there was much less knowledge about the t- and b-cells. The concept of over-activating seems to be coherent with some "weakening" of the innate immune system; that weakening being compensated by activation of the adaptive immune system. This might be some interesting theory, which, however, the publishers clearly did not want to convey).
Again, fair interpretation of the textbook list entry pretty much depends on what is to be understood by "immune system" in that specific context: clearly, cortisol dampens inflammation, conversely, hyposecretion should lead to hyper-inflammation. Now, for a fair interpretation of the saying one might separate "immune reaction causing inflammatory symptoms" from "the" immune system in the sense of the statement, raised by cortisol, known for dampening those inflammatory reactions of "the other system" (causing inflammatory symptoms, Cox etc). Then, they may say: the less cortisol, the more weakening of the "cortisol immune system". That would not even be pleonasm as there must be some opposite way of stating "cortisol fortifies the immune system".
