Cheap date is slang for someone who has a very low level of alcohol tolerance and requires only a small amount of alcohol to get tipsy and/or drunk. Therefore, this person doesn't need to buy (or have their date buy) as much alcohol to feel the effects, hence making them a cheap date.
Cheap date is also the name of a specific mutation of a gene found in the common fruit fly (Drosophila melanogaster). It was discovered in 1998 by a group of researchers at the University of California, San Francisco. The cheap date mutation inactivates the gene, which makes the flies much more susceptible to alcohol. Apparently this is measured by monitoring the activity of the flies after they have been exposed to ethanol vapors. When inebriated, the flies are highly active and uncoordinated and eventually become dazed and pass out just like humans.
Interestingly, the cheap date mutation is an allele of a gene called "amnesiac," which is thought to be important for memory. This gene encodes a neuropeptide that activates the cAMP signaling pathway and increases the levels of cAMP in the organism. cAMP levels are thought to be directly related to alcohol tolerance; higher levels tend to give an organism a higher tolerance. Therefore, flies that have the cheap date mutation produce less cAMP because the amnesiac gene is inactivated and are much more susceptible to the effects of alcohol.
Who said scientists don't have a sense of humor?
Abstract of the original paper: http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=9635429&dopt=Abstract