Assuming you've decided to invest in the gro-lights, containers, and nutrient mixes, we'd like to recommend some hydroponic vegetables you could try growing.
The following are plant types you might want to try under hydroponic cultivation, with some tips for best results. Many can be had in special greenhouse and forcing varieties. Some of these are available as specially adapted "hydroponic vegetables" bred to grow independent of fluctuations in day length (or photoperiod, which is constant in hydroponics under lights) that schedules the ripening cycles of most plants to fit the seasons of the year.
Artichoke. A kind of giant thistle that requires a large planter and only produces one large and two or three smaller side-shoot buds. A frost-tender perennial that takes several years to mature. Needs extra potassium in its diet.
Asparagus. Super-hardy perennial that is best grown outdoors unless you don't have an outdoors. Can be raised in two-foot-wide-and deep planting boxes filled with a cheap local medium such as gravel. Set well-rimmed asparagus roots a foot deep and two feet apart in cells of Cocopeat amid the rocks. Rig a trickle-tube or drip ring to the center of each root. You can companion-plant a whole variety of plants growing in pots or "grocubes" around the top of the bed. Try strawberries and horseradish.
Beans. All kinds and maturities of fresh beans (snap, lima, soy, fava) will do well. Like all big seeds, they are best started in Cocopeat or perlite mix; if pushed down into a rockwool cube, the seeds may rot. Bush beans are easiest and fastest, but pole beans can be grown on string supports. Shift from a general-purpose nutrient to a low nitrogen solution once the plants have finished leafy growth.
Beets for greens and roots. Best in a box filled with a loose medium such as a sand, perlite, or vermiculite mix.
Broccoli. Raab-type leafy variety grows fast and can be harvested over several weeks. Heading types need a lot of space. Varieties such as old-time De Cicco with small main heads and lots of side sprouts produce the most in the least space. Feed heavily once heading begins.
Carrots. Same as for beets. The thimble-size miniatures do well. Long types need a deep medium bed.
Celery. As hard to grow in hydroponics as in soil. Needs a light medium, uninterrupted moisture (three drenchings a day), and cool soil temperatures.
Cucumbers. Try the greenhouse varieties. You must provide support for vines. Best grown in cooler, dryer seasons to avoid mildew plagues that are hard to prevent in hot humid conditions (mildew spores are in the air everywhere). It's hard not to spread molds, as plants must be suckered (sprouts at base of fruiting stems nipped out). Gynoecious (all female) varieties need special care; instructions come with the very expensive seeds (up to $1 per seed available in books. A hydroponics special; there is much information available on growing cucumbers. Since no ripening time is needed, you can harvest edible fruit as' quickly as 40 days after seedling. Growers in cooler areas should try the Corona variety from Stokes Seeds.
Eggplant. Can be grown as a perennial and will grow into a woody bush that blossoms and fruits several times a year — size of plant limited only by amount of medium. You might choose dwarf-growing varieties. When fruit begins to develop, reduce nitrogen. Pollinate blossoms as with tomatoes.
Flowers. Nearly all do well ... from long-stemmed roses and orchids that require special care to vining nasturtiums that will grow wild through your other plants and contribute young leaves and spicy flowers (without aphids) to salads.
Herbs. Lush herbs such as basil thrive. Arid land aromatics such as thyme do better outside in the rock garden. We are told that "extralegal" exotics, including certain varieties of cactus, mushroom, poppy, and rope fiber, are cultivated under lights in many rural locales. But we also hear that the DEA (federal narcs) monitors sales of hydroponics gear and likes to bug or tail large shipments of FloraGro or Hi-Intensity lamps going by UPS to some isolated, rundown old farmhouse in the country. The Internet abounds with information and equipment for indoor exotic-herb growing. For example, you can find a fully-enclosed self-contained growing chamber six feet high and over a yard through. It costs $2,000 and you'd have a hard time convincing the SWAT team leader that you want it to grow sweet corn in January. MOTHER EARTH NEWS suggests you stick with tomatoes and cucumbers.
Lettuce, spinach, salad greens. Keep roots cool. Leaf lettuces grow so fast you can see a daily difference. Use a high nitrogen solution and a good-draining medium. They are best suited to flowing water systems.
Melons and summer squash. They need a good draining medium and low humidity to prevent mildew. Dwarf-growing varieties take up less space. You can trellis vines if developing fruit are given individual support. Keep pH near neutral. Intolerant of acidic media.
Green onions, scallions. Start from seed, not sets. Can be grown almost touching. Mild flavor, succulent.
Peas. Edible-pod sugar peas are most rewarding. Grow on string trellis with a high nitrogen solution.
Peppers. Grow the same as tomatoes. Some of the hot varieties will become perennial bushes if you let them.
Sweet potatoes. Need ample root space. Best grown in large bins filled with a coarse sand medium. The attractive vines can be trellised. For hydroponic color, intermingle with morning glories and scarlet runner beans on a string-trellis against one wall.
Strawberries. Irresistible. Wash thoroughly, dip in mild Clorox solution, rinse immediately, and plant fully-dormant (no sprouts) bare-rooted Young plants a foot apart each way in pots or bags of medium. Feed with drip tubes to each plant. Nip off runners, rooting the best to grow for next year's crop. Hand-pollinate flowers. You can prune all but the best developed young fruit on a plant and produce perfect fruit almost the size of lemons. Ideal for planting in space-saving towers and growing in the corner of a city apartment.
Potatoes. Seldom grown in hydroponics. But you might try fingerlings or another gourmet variety in a bin that can hold four cubic feet of light medium such as Cocopeat. Needs its own nutrient solution, heavy in P and K, and maintained at a mildly acidic pH of 6.0.
Tomatoes. Like cucumbers, a hydroponics special. Whole books are available. You can have ripe tomatoes in two months less growing time than outdoors. Remove all suckers as they sprout. Trellis or stake vines to increase exposure to light and air flow that will help keep molds in check. It is best to grow small or medium-fruited greenhouse forcing varieties. Cobra, a French beefsteak variety, will ripen all the way through and taste like a real tomato unlike many hothouse varieties. Cencara, also French and from Stokes, is an oval cluster-type. Clusters should be pruned to five fruit. Needs a high concentration of food with extra potassium while fruit develop.