The fish and plants you select for your aquaponic system should have similar needs as far as temperature and pH. There will always be some compromise to the needs of the fish and plants but, the closer they match, the more success you will have.
As a general rule, warm, fresh water, fish and leafy crops such as lettuce and herbs will do the best. In a system heavily stocked with fish, you may have luck with fruiting plants such as tomatoes and peppers.
Fish that we have raised in aquaponics with good results:
tilapia
blue gill/brim
sunfish
crappie
koi
fancy goldfish
pacu
various ornamental fish such as angelfish, guppies, tetras, swordfish, mollies
Other fish raised in aquaponics:
carp
barramundi
silver perch, golden perch
yellow perch
Catfish
Large mouth Bass
Plants that will do well in any aquaponic system:
any leafy lettuce
pak choi
kale
swiss chard
arugula
basil
mint
watercress
chives
most common house plants
Plants that have higher nutritional demands and will only do well in a heavily stocked, well established aquaponic system:
tomatoes
peppers
cucumbers
beans
peas
squash
broccoli
cauliflower
cabbage
These are of the other crops that have been grown in aquaponic systems:

bananas
dwarf citrus trees: lemons, limes and oranges
dwarf pomegranate tree
sweet corn
micro greens
beets
radishes
carrots
onions
edible flowers: nasturtium, violas, orchids