I looked at the Material Safety Datasheet for ammonium persulfate, and all it said about disposal was not to let into natural water supplies, dispose in accordance with local regulations.
So where do I go to find out about the regulations? The city, county, state?
i would recommend that you take it down to the local tip and ask someone there. if you are careful you may just get away with flushing it away because all that sewerage has to be treated and it may be caught out there(but i wouldn't test it). i recomend that you either go to an electronics shop, your local council or the tip.
If you have a university or school near you, you could take it there, and ask them to put it in their "chemical waste" container. Any chemistry lab will have a container for aqueous waste.
I'd suggest putting a few pcs of copper in it to let it all react. Then put it in an old plastic soda bottle and throw it in the trash.
Dumping it down the toilet is no good for your pipes.
A trash dump may not be the best place for it, but it certainly won't be the worst thing there. If you are really worried about it, see if your town has a toxic waste day, where you can bring paint and such to a designated area to be disposed of.
Holy crap don't put it in a water bottle and leave it in the trash. That is a terrible idea. Ammonium persulafe with the copper from your board is a toxic liquid. Dispose of it properly. Find a hazardous waste disposal site. Phasor is right in that Universities are a good place to start.
I just throw it in our pool. Copper is used as an algaecide and the amount added is hundred of times more than any board etchant I might use.
If you want to be extra safe then mix in sodium bicarbonate (used in baking - ask mum) until it stops fizzing and the copper will precipitate as copper carbonate. You can throw the copper free solution down the drain and the blue powder (sludge) in the trash as it is now in an insoluble form. It will take a while for the powder to settle or you can pour it though a coffee filter.
The first time you said this (in another thread,) I really thought you were joking, but then I read an article on regenerating etchant, or on pool maintenance - can't remember which - and realized that you weren't.