I'm aware that cyanobacteria are responsible for the Great Oxidation Event and are very effective at turning CO2 into oxygen so it stands to reason that with the proper nutrients for the bacteria, you could make a more effective life support system. Are there any reasons why this would/wouldn't work and why it hasn't been implemented?
Why this wouldn't work? It works for the Earth; the reason why it is not implemented in space is purely in the engineering limitations.
- Cyanobacteria live in water, humans live in air.
Gravity is good at separating water from air, leaving a surface for the gas exchange. Microgravity is very good at mixing everything, so we should think about another approach for obtaining waterless air for humans and (mostly) airless water for cyanobacteria.
Light. Cyanobacteria are moderately efficient at producing oxygen. One will need few sq. meters per astronaut of transparent (but otherwise gastight) windows in order to allow enough light in. Not a trivial task in space. You will have to orient the windows in a favorable direction or make an extra amount of them. Also, good luck cooling the whole thing against these 1.5kW per square meter of heat input.
Closed carbon/nitrogen/phosphorus/etc cycle. Cyanobacteria don't only produce oxygen. They also reproduce at a rather high rate. One will want to recycle some of this biomass (it is barely edible and you have to extract it from the water in the first place).
In short, possible, but the transport to/from the Earth is WAY cheaper (for now).
Another engineering aspect is controllability. Most electrical/mechanical/chemical life support systems can be throttled to anywhere between 0% and 100% of production capacity within minutes, if not seconds. In particular you can shut the system down for repairs and also carry a fully inert system as a cold spare. And since the atmosphere inside a spacecraft is far from a natural equilibrium, it is really important to keep it at a stable survivable level all the time. You definitely do not want a "Great Oxidation Event" to occur aboard and oxygen levels dropping to low are of equal danger.
With biological systems, the same level of control is much harder. Changing oxygen production is much more indirect, via nutrients and light influx and even though bacteria reproduce fast, significant changes to the population will take hours or even days. And if by some accident you manage kill your bacteria by contaminating the system, getting it back to running will take a long time.
I am not saying, that it is not possible to build such a system, but it is a challenge and probably too dangerous without having a conventional system as a backup along with a few weeks supply. So for all the missions that are done currently, using the conventional system directly might be the wiser choice.
Environmental Control, and Life Support Systems need to be tailored to their application. There are many tradeoffs such as how much they weigh, how complex they are, how reliable they are and how efficient they are.
As a general rule the longer the duration of the mission, the greater the gains from life support closure. So missions of a few days can get by with little recycling whereas missions of many years need very high degrees of closure.
Any cyanobacteria oxygen recycling system would be too heavy for the short duration missions. But although it might be considered for long duration missions like a Mars base there are probably better materials to use. For example edible crops which can help further close the recycling loop. What happens to the dead/excess cyanobacteria?
In light of a very recent news item I thought I'd turn my comments into answer so the information isn't lost should comments be deleted.
Having visited a mineral processing plant that used bacteria to treat ground up minerals from a mine, two things struck me about plants that rely on bacteria to do useful work. The first is the bacteria usually need a narrow temperature range to optimally do what is required of them. The second, is sometimes bacteria can produce very unpleasant odors in the natural course of what they do. Any such odors that may be produced by cyanobacteria would need to be dealt with so the crew were unaffected.
Another reason to be careful about using cyanobacteria, from Doctors investigate mystery brain disease in Canada.
... beta-methylamino-L-alanine (BMAA) - which has been implicated as an environmental risk in the development of diseases like Alzheimer's and Parkinson's.
BMAA is produced by cyanobacteria, commonly known as blue-green algae.
Not exposing a crew to BMAA and its possible neuro-degenerative effects is another reason not to have life support system based on cyanobacteria.