What plant produces the most oxygen?

Phytoplankton (a plant) in the oceans produce 70% of the world's oxygen, by far the largest contribution from any type of plant.

If you want the individual plant that gives off the most oxygen each year, it is probably the 43 hectare (106 acre) network of quaking aspen tree trunks (Populus tremuloides) in Utah that the Guinness Book of World Records lists as the tree world's largest living organism. The mass of the quaking aspen is estimated at 6 million kg (6,600 tons). The quaking aspen is considered a single organism because all the stems originate from one root system. There was an article in Discover magazine several years ago on the world record quaking aspen.

In terms of the most oxygen produced per unit ground area covered by a single plant species, it would be a tropical grass or other species with C-4 photosynthesis, such as cultivated corn (Zea mays). For natural ecosystems per unit ground area, tropical rain forests generally have the highest annual rates of photosynthesis.

Reference

Salisbury, F.B. and Ross, C.W. 1985. Plant Physiology. Belmont, CA: Wadsworth Publishing.

If you simply want to place plants in your house you need to look at getting rid of all pollutants, NOT just increasing oxygen levels.

These are the top pollutant reducing plants as discovered during a 2 year experiment by NASA.

TOP 10 plants most effective in removing:
Formaldehyde, Benzene, and Carbon Monoxide from the air

Bamboo Palm - Chamaedorea Seifritzii

Chinese Evergreen - Aglaonema Modestum

English Ivy Hedera Helix

Gerbera Daisy Gerbera Jamesonii

Janet Craig - Dracaena "Janet Craig"

Marginata - Dracaena Marginata

Mass cane/Corn Plant - Dracaena Massangeana

Mother-in-Law's Tongue Sansevieria Laurentii

Pot Mum - Chrysantheium morifolium

Peace Lily - Spathiphyllum

Warneckii - Dracaena "Warneckii"