They are talking about lithium recovery, but there is a less exotic byproduct I'm interested in. One tonne (≈ 1 m^3) of seawater contains about 1.3 kilograms of magnesium, equivalent to about 4 kg of magnesite ore. Typical desal prices are on the order of $1 per tonne. Magnesite ore goes for about $100 per tonne, so the crude magnesium in a tonne of seawater is worth about $0.40, which could account for a substantial fraction of the desalination cost. (These numbers are very rough.)

Now you ask: why don't we just recover magnesium from brines if it's so great? Magnesium recovery from seawater isn't that easy: typically you have to treat it with some kind of alkali (often Ca(OH)2), so the cost is dominated by the extraction process (your alkali is consumed!), and you're competing with a pretty cheap ore. But if you have a solid byproduct, instead of a liquid, the options for magnesium recovery might be a lot more efficient, potentially offsetting the cost.

The fourth-most-prevalent ion, sulfate, might also be interesting, at least in a hypothetical post-petroleum future where sulfur as a byproduct of fossil fuel extraction is no longer "free". Sulfate is also annoying to extract from seawater, but again if we have a solid, the rules change.

As for the "table" salt itself, I think we'd quickly saturate (!) the market.