I was just reading this booklet about using duckweed to purify greywater. (The pages jump all over the place so following the flow is a bit of a challenge.) It sounds like duckweed thrives in that environment. As long as no toxic detergents were used, do you think it could be harvested and used as fish food?
Well, issue is if any soaps or products are used that could hurt the fish, I don't see any good easy way to make sure those products/soaps won't get into the aquaponic system when feeding the fish the duckweed.
Duckweed can be rather useful stuff but you have to remember that it is very much a product of how it's grown and if feeding it fresh and wet, you better make sure whatever is used to grow the duckweed is something you feel safe putting into your aquaponics system and remember you plan to eat food from that aquaponics system too.
The following article would warm me that the duckweed grown in grey water would likely contain lots of un-desireable elements. I would look to our chemistry savy friends for answers.
Arsenic (As) and cadmium (Cd) pollution in water is an important global issue. Phytofiltration is an eco-friendly technology that helps clean up pollutants using ornamental plants, such as Micranthemum umbrosum (J.F. Gmel) S.F. Blake. After a seven-day hydroponic experiment, M. umbrosum removed 79.3–89.5% As and 60–73.1% Cd from 0 to 1.0 μg As mL–1 and 0.3 to 30.0 μg Cd mL–1 solutions, respectively. For As treatment, root to stem and stem to leaf translocation factors greater than 1.0 indicated that accumulation of As in leaves was large compared to that in stem and roots. However, the accumulation of Cd in roots was higher than that in the leaves and stem. In addition, M. umbrosum completely removed Cd within three days from 0.38 to around 0 μg mL–1Cd in the solution when the plant was exchanged daily. Bio-concentration factors (2350 for As and 3027 for Cd) for M. umbrosum were higher than for other As and Cd phytoremediators. The results show that M. umbrosum can be an effective accumulator of Cd and a hyper-accumulator of As, as it can lower As toxicity to a level close to the limit recommended by the World Health Organization (0.01 μg As mL–1).
If one were careful to make sure that their grey water never contained anything that could detrimentally bio-accumulate, then I suppose it could be ok.
However, in our modern day and age, making sure no modern cleaners, chemicals, beauty products, hair care products, etc ever get used on the property might be very difficult. And old fashion soaps would be dangerous to fish. So that doesn't leave much.