I am not clear on what you mean by "no net effect", and not entirely sure how tears are comparable with blood since tears are expelled from the body; blood (and any sodium it carries away from the brain) hopefully remains in the body (not to mention that blood contains a lot more than just NaCl, compared with tears). I am also curious as to how quickly the hypothalamus would respond to a local drop in sodium ions. What you say is plausible; I do tend to agree with you about the study interpretation. Like you, I really want to hear from an expert in this field.
EDIT to reply to your edit: No, my edit 2 above was made before I saw any replies, sorry about that. Yes, it is certainly nothing but speculation.
(Thank you for pointing me in the direction of further study - I'm afraid this type of thing is a bit beyond my education when it comes to biology)
"No net effect" = tears are produced by the lacrimal glad. I am more of an interested hack, compared to someone that actually knows something (medical professional like a neurologist or ophthalmologist), but my educated guess is that cells in the lacrimal gland produce tears entirely from inputs from the circulatory system, and because of the circulatory system structure in your brain v. face, it's actually rather not as local as the physiological proximity suggests. So the change would be indirect, mediated by blood, and since the concentration of sodium is the same in each crying wouldn't effect blood sodium concentration in the first place. You may under-appreciate how rapid blood moves through the body, too, and quickly any local change would move to equilibrium.
ps: you're welcome, happy studying! (it's daunting, keep in mind https://news.ycombinator.com/item?id=1390027, and remember that some of the smartest people out there become doctors, and they spend a dozen+ years training in just single areas of study and indefinite continuing education)
EDIT to reply to your edit: No, my edit 2 above was made before I saw any replies, sorry about that. Yes, it is certainly nothing but speculation.
(Thank you for pointing me in the direction of further study - I'm afraid this type of thing is a bit beyond my education when it comes to biology)