Vitamin C, or ascorbic acid, is synthesized from a hexose in plants. The hexose can be glucose, fructose, or mannose. All those sugars come from photosynthesis. You might be able to improve photosynthesis by adding potassium and magnesium, two ions that are crucial in photosynthesis. However, the ascorbate biosynthesis pathway is under negative control so that the increase concentration of ascorbate would inhibit its own production. Plant uses this to make sure they do not have too much vitamin C. In fact ascorbate controls the expression of all 11 enzymes in the pathway ( source . )

Nothing short of selective breeding or genetic engineering would help. However, all hope is not lost.

Ascorbate also promote the degradation of GDP-mannose pyrophosphorylase (GMP), an enzyme involved in the ascorbate biosynthesis. Kiwi fruit is rich in vitamin C, and known to have high expression level of GMP. GMP is first marked by the COP9 complex and then degraded by the 26S proteasome. Perhaps the addition of COP9 inhibitor can maintain the level of GMP and increase vitamin C yield. A research showed that zinc dithiocarbamate can inhibit COP9 in animals source. ). If tomato can uptake zinc dithiocarbamate from the media, then GMP level could be maintained despite the increase ascorbate concentration. If GMP level is maintained, then the ascorbate concentration should be higher comparing to the control.

It should always be kept in mind that whenever you handle chemicals, you must read the corresponding safety data sheets so you can handle them safely. Zine dithiocarbamate is a skin irritant. Please wear appropriate personal safety equipment when handling it and follow the safety protocol.

Yes, there has been at least one study that examined the levels of potassium (K), phosphorous (P), calcium (Ca), Iron (Fe), and vitamin C in tomatoes grown in vermicompost (aka worm castings or what worms poop out when they eat your leftovers) versus hydroponic media. Their study found that Ca and vitamin C levels were higher in the vermicompost tomatoes than in the hydroponic tomatoes. The full article can be found for downloading here. I would be surprised if plant scientists weren't studying how to make more nutrient-rich hydroponic tomatoes. Unfortunately, it may not be as easy as simply adding more calcium or vitamin C to the water! They would have to find a way to introduce these molecules so that the plants would want to take them up and keep them active, or to promote vitamin C production within the plants themselves.

Regarding tomatoes in particular, this article has some sections indicating that optimizing the hydroponic growing conditions can enhance levels of several nutrients, including vit. C.

Going the other way, if the solution is un-optimized, then the quality of the vegetables will suffer. Alternatively, having optimal soil conditions can also produce a better, more nutritious crop than one grown in "whatever the weather was that year."

A more direct article is this with lettuce, which showed that the hydroponically grown crops have as much or more vitamin C (aka ascorbic acid) compared to soil grown crops. Their nutrient solution was not altered after purchase to try to increase the amount of any vitamins, but the nutrition content is proprietary and the manufacturer may try for "enhanced" nutrition (my comments, not theirs, and see later section on why they might not). The authors do indicate though, that the greater vitamin C content can be due in part to the difference in growing conditions - the temperature, sunlight, etc. varied between the locations in the greenhouse, and these are known to impact the nutrient content of vegetables.

As far as modifying of hydroponic solutions to improve nutrition, not really, as long as the nutrients are already being provided in the correct amounts. First, different crops require different conditions, so hydroponic solutions may be formulated for a specific crop (for example strawberries or lettuce.

Using the "wrong" hydroponic solution will give less nutritious tomatoes. Although one might consider increasing nutrients provided to the plants to increase vitamin C, as described here (Nutrient Antagonism section), using too much of one growth nutrient can worsen the growing conditions. That doesn't mean that nothing is being done though - there is research on the effect of nutrient content of hydroponic solutions on the nutritional value of crops. However, increasing the nutrients beyond those current levels will likely require genetic changes to the plant so that they are capable of "manufacturing" more vitamins, etc. during their growth span.