Neem and pyerthrin both have their applications, and both are actually organic (assuming the extraction process falls under guidelines for organic production).
Neem is the best organic PREVENTATIVE I can think of for many problems, but has a very foul odor and should not be applied directly to the buds themselves. Coriander oil and citrus oil products are somewhat less effective, but similar, and are less gross smelling. I'd switch to one of these within 3 weeks of harvest.
Pyrethrin (derived from a species in the chrysanthemum family) is an instant knock-down insecticide that breaks down in just days in intense light. You can apply it directly to the buds IF it has no activators like piperonyl butoxide in it. Don't do this less than 2-3 weeks before harvest, though. However, I would avoid doing that, as it's usually dissolved in water and water on buds = bad. Use it on the leaves, and for fighting bugs late in flower, try a pure pyrethrin room fogger/bomb like Doktor Doom.
Or skip the pesticides and release some ladybugs. They are the #1 predator for dealing with an aphid problem.

How the heck you get aphids in there to begin with? Got some tomato or pepper plants inside your house? Usually aphids leave weed alone indoors.