Cousin Eddie is drinking Meister Brau, a popular beer brewed in Chicago for much of the 20th century.
Meister Brau, and other beers from the brewery of the same name, were sold to Miller in the early-70s, so Cousin Eddie is actually drinking a predecessor of Miller Lite.
Cousin Eddie can often be found with a can of Meister Brau in his hand. The can can clearly be seen in the scene where Eddie empties the septic tank in Clark’s yard.
Meister Brau was brewed by Peter Hand Brewing Company in Chicago, Illinois from 1899 to 1978 when it was sold to Miller. Peter Hand was a Prussian immigrant and veteran of the American civil war, and he opened the brewery in 1891.
The brewery was forced to close for 13 years due to prohibition from 1920 to 1933, but was able to last and even continue to expand after reopening.
They invested heavily in the Chicago market, sponsoring many professional sports teams. They saw great success with the launch of Meister Brau Lite, which became incredibly popular.
Though Meister Brau proper is no longer being manufactured, curious fans will be glad to know that, following the sale to Miller, the beer was rebranded to Miller Lite.