I am a first year medical student and my textbook of biochemistry under the heading of sodium under the section minerals barely touched upon hyponatremia...
You are too impatient, everything will clear up in the following years, the first year of medical studies provides you with infrastructure, you will eventually be taught what this is about. Medical biochemistry as a subject is not deeply concerned with pathophysiology, so there is no need to expand upon subjects of general pathology or internal medicine to a level exceeding that of simply providing some motivation.
Hyponatremia is a result of, primarily, an imbalance between total body sodium and total body water (volume status). Multiple mechanisms are at play, it's not that simple. In general, when volume status is "normal" while hyponatremia manifests, it is called euvolemic. How can that be? Well, homeostatic dysregulation leads to more water retainment than appropriate; not enough to cause volume overload, but definitely enough to dilute total sodium and reduce the concentration.
The Long Version
You have absolutely missed the point of what actually is hyponatremia, in your question. There is no intuition behind it, really. Blood sodium concentration is tightly controlled and when it's under the lower expected (a pretty vague term, I know) limit, then you have hyponatremia. To quote from :
Hyponatremia (serum sodium [SNa] <136 mmol/L) is a common water balance disorder that often poses a diagnostic or therapeutic challenge.
That's it. As you see, it is all about a balance of sodium and total fluid. Too much or too little of either, and you have an imbalance. Too much fluid increases the denominator of the fraction (mmol/L, remember?) and, thus, the fraction decreases in absolute value. Too little sodium, and, again, the absolute value of the fraction decreases. In fact, both the total sodium and the fluid content vary at the same time, therefore there is more than a dozen ways, in which an imbalance can manifest.
The intuition you are probably looking for is most likely related to your understanding of the term, which may be misleading for the uninitiated. Hyponatremia, in Greek, stands for low sodium in the blood, but the word does not really explain if it's talking about concentration or total content (two very different things). In a medical context, it is about concentration, and this is a little bit baffling, because you can have hyponatremia even with increased total body sodium, just because the total body fluids may have expanded disproportionately much, so the fraction now has a larger numerator, but a far larger denominator, so the numerator increase is not enough to counter that.
Since hyponatremia is all about concentration, it is natural to expect that we would like to parameterise it based on the two parameters it is based on: sodium content and water content.
Because the sodium concentration is a ratio, we choose to talk only about one of the parameters, and it will be evident (sort of, like, left as an exercise to the learner) what the change in the other parameter will be. The parameter we used for classification is the total fluid volume, hence hypo-, eu- and hyper-volemic hyponatremia.
Hyponatremia means that the fraction decreases, so:
- Hypovolemic, means that the total body fluids did decrease substantially but the total sodium decreased much more so, thereby reducing the total fraction. As the MSD Manual puts it:
Deficiencies in both total body water and total body sodium exist, although proportionally more sodium than water has been lost; the sodium deficit causes hypovolemia.
- Euvolemic means that the total body fluids are within normal range, but the total sodium content is relatively less than what would be needed to maintain the expected normal levels. Confer the corresponding part from the MSD Manual:
In euvolemic (dilutional) hyponatremia, total body sodium and thus ECF volume are normal or near-normal; however, TBW* is increased.
- Hypervolemic means that the total body fluids increased substantially, and the total sodium content may have decreased, remained constant, or even increased, but not adequately so, as to avoid a substantial reduction of the fraction. Again, here is a fair explanation from the MSD Manual:
Hypervolemic hyponatremia is characterized by an increase in both total body sodium (and thus ECF volume) and total body water with a relatively greater increase in TBW*.
*TBW: Total Body Weight
See how this goes? Hypo-, eu- and hyper-volemic states the total volume status as an assessment, it is not related to the sodium concentration per se. Therefore, you get two pieces of information in one by using this characterisation with hyponatremia.
Finally, the reason behind this classification is that hyponatremia is absolutely relevant to the total volume status of a patient.
There is a better way to look at hyponatremia, and that is to regard the intravascular fluid tonicity. True hyponatremia occurs when the intravascular fluid is hypotonic as a result of the hyponatremia. False hyponatremia is the observation of hyponatremia as a secondary consequence of a different process. To better help you understand why this is termed false, take a look at what hyponatremia causes (as opposed to what causes hyponatremia). Quoting Wikipedia:
When sodium levels in the blood become very low, water enters the brain cells and causes them to swell (cerebral edema).
Water enters brain cells not because of the low sodium levels, but because of the low intravascular fluid tonicity. If too much glucose, or the existence of mannitol, draw water into the blood, that is because they increase blood tonicity, thus rendering it hypertonic. In that setting, the extra water drawn into the blood does decrease the sodium concentration, but there are no "hyponatremic" effects, no water enters the brain cells, because the intravascular fluid tonicity is not low. How would you call a hyponatremia that does not produce any signs or symptoms (because, obviously, the main mechanism is not at play)? In wondering about this, make sure to understand how "hyponatremia" is tightly related to the role of sodium as a maintainer of blood oncotic pressure.
I think all that rant should be enough to get you started. I know that you probably have more questions than answers at this point, but this is, more or less, what the first years of medicine are like. Eventually, however, and with study and perseverance, things should fall into place.
 Hoorn EJ, Zietse R. Diagnosis and Treatment of Hyponatremia: Compilation of the Guidelines. J Am Soc Nephrol. 2017 May;28(5):1340-1349. doi: 10.1681/ASN.2016101139. Epub 2017 Feb 7. PMID: 28174217; PMCID: PMC5407738.