List of beneficial weeds

This is an incomplete list, which may never be able to satisfy certain standards for completeness.

Revisions and sourced additions are welcome.

This is a list of undomesticated or feral plants, considered weeds, yet having some positive effects or uses, often being ideal as companion plants in gardens.

Beneficial weeds fall into several categories:

Pest-repellant

Crow garlic -- a wild allium which repels certain insect pests and is edible
Cocklebur -- repels armyworms
Goldenrod -- repels some bad insects, and shelters several useful predatory species
Milkweed -- repels wireworms
Caper Spurge -- believed to repel moles

Edible

Cornflower flowers are pretty little baskets, of various colors, which can be served as edible garnish to decorate salads.
Painter's brush weed
Chickweed -- good chopped, in salad, also good ground cover
Burdock -- roots are edible
Lambsquarters -- leaves and shoots, raw, also prevents erosion, also distracts leaf miners from nearby crops
Shepherd's purse -- blanch and sauté the leaves

"Don't use shepherd's purse if you're pregnant or breast-feeding. Use this herb cautiously, if at all, if you have heart or lung disease. Use shepherd's purse cautiously if you take a drug that alters your heart rate or depresses your nervous system" Quote from following site http://www.womens-health-club.com/herbs/shepherds-purse.htm

Stinging nettle -- High nutritional value. Wear gloves while picking the leaves. Use in soups, etc. as you would with spinach.
Purslane -- prepared raw for salads or sautéed
Watercress -- can be eaten raw or cooked; is considered a weed in some cultures
Dandelion -- flowers can be used to make wine; leaves are edible and good for digestion; roots sometimes used as coffee substitute

Provides habitat for beneficial insects

clover -- attracts predatory insects, also good for soil
Solanum -- provides cover for predatory ground beetles which hunt aphids
Pigweed / Amaranthus -- also shelters ground beetles, breaks up hard soil, allowing other plants to develop deeper roots
Queen Anne's lace -- attracts predatory insects like lacewings, its seeds contain estrogen and are used in folk/herbal medicine as a contraceptive, and its root breaks up hard soil/deadpan. This plant is also the ancestor of the cultivate carrot; in fact it's the same species, and carrots will gradually return to this state if left to breed in the wild.
wild blackberry -- attracts predatory insects, and produces berries
motherwort -- attracts bees
wild mustard -- protects predatory insects
Joe-Pye weed habitat for pollinators and predatory insects

Provides shelter

normal grass can be used as ground cover, especially when very fertile soil may otherwise provide too much nitrogen to plants like tomatoes and peppers, which will then stay in leaf-growing mode and fail to produce fruit
Purslane -- can be used to protect soil from erosion

Trap Crops

Trap crops draw potential pests away from the actual crop intended for cultivation

Multiflora Rose -- distracts Japanese beetles from good crops (This is a non-native invasive species in North America - see link )
Nasturtium -- attracts caterpillars and aphids, so planting them alongside or around vegetables such as lettuce or cabbage will protect them, as the egg-laying insects will tend to prefer the nasturtium.

Medicinal use

Bashful mimosa -- various herbalist uses
Rumex -- Dock, which commonly grows in association with nettle, is rumoured to cure or ease their sting. Crush a leaf before applying to affected area.

Other

Cannabis -- clothes can be made out of hemp, as well as a form of paper both cheaper than and superior to wood-pulp paper. The seeds can also be used in most of the same capacities as soybeans, both for food and as a source of vegetable oil and fuel alcohol.
Dandelion -- Breaks up dense soil, helping vegetable roots go deeper. If picked while in season, leaves and flowers are edible as a salad component. Repels armyworms
Nightshade -- breaks up hardpan, allowing roots to grow deeper
wild vetch the early cousin of the wondrous (and expensive) cover crop hairy vetch.

References:

Peterson, L.A. & Peterson, R.T. (1999). A Field Guide to Edible Wild Plants: Eastern and central North America. Houghton-Mifflin.

Duke, J.A., Foster, S., & Peterson, R.T. (1999). A Field Guide to Medicinal Plants and Herbs: Of Eastern and Central North America. Houghton-Mifflin.

Gibbon, E. (1988). Stalking the Wild Asparagus. Alan C. Hood & Company .