What does the gene name "lexA" stand for?
I'd go with Lambda Excision A. Terms like lex or rec often stand for what'd be termed a mnemonic, where for example rec may stand for recombination, or umu for UV mutator. The naming conventions can be difficult.

Edit: A 1981 study by Roger Brent and Mark Ptashne notes some data from initial studies that showed the lexA repressor downregulated a number of genes including himA, which they note to be required for integration of lambda phage.

http://www.pnas.org/content/78/7/4204.full.pdf