Lake Eyre in Australia has an area of roughly 4,000 square miles, but the basin itself is over 450,000 square miles. If the entire basin were freshwater, how would that affect the Outback's climate?
It's hard to say exactly how a randomly placed large body of water will effect the area's climate. In the long run, GrandmasterB is most likely correct. In the mean time, you will not only change your local climate, but that of a much larger area.
Water has an incredibly high heat capacity. That means that it can store a lot of heat. Essentially, the water will absorb a lot of the heat that would normally be put into the air during the day, and will then slowly release it during the night. Now rather than have really hot days, and really cold nights, I would assume the outback has a more moderate climate.
If we assume there is a way for the lake to be recharged (otherwise, as noted, it will eventually dry up and revert to the current size), then the presence of a large body of water will do two things:
Moderate the local microclimate. The water will absorb the heat of the day and slowly release it at night, reducing the temperature swings. Over the longer term, it will also tend to reduce the overall temperature during the summer and increase it during the winter (we are talking a very small overall change; there won't be palm lined beaches in winter...)
Affect the larger hydrological cycle. The evaporation of water from the lake will result in a large area receiving rain in the "downwind" area. Given the size of Australia, there is some risk that the evaporated water will actually be deposited outside the boundaries of Australia itself (i.e. the rain falls in the oceans), which would make this a bit pointless as a geoengineering project.
Like all closed basins, the greatest danger is the lake itself will gradually become briny and toxic, since as the water flows in, the sediments and accumulated "stuff" is deposited but cannot leave. As the water evaporates, what is left becomes saltier, and toxins become more concentrated, much like the situation in the Salton Sea in the US or the Aral Sea in central Asia.