It is an important gene expressed in E. coli that represses the SOS response and also the expression of lambda lytic phase genes. UV light and damage to DNA is responsible for its breakdown and hence the expression of repair genes and, if present, lambda phage genes. However I couldn't find a reference for its original name.
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.
According to this website, it is a mnemonic for "lambda excision". I have also found this usage in scientific literature (eg. Harami et al., 2013). However, neither of these sources reference anything and I cannot find any defining paper. Almost all papers simply refer to it as lexA.
As far as I can find (it's difficult to dig up these old papers), this is how the locus (not necessarily gene) was originally described (Howard-Flanders and Boyce, 1966):
There is a ... locus, lex (locus for X-ray sensitivity), in E. coli K-12 ... [A] mutation at this locus increases the extent of UV-induced DNA breakdown. This effect is similar to, but less extreme than, that of the rec locus ...
It's possible that as this locus was better characterised, the name lex was retained for consistency with previous literature but redefined to reflect a better understanding of its function (although I don't think that "lambda excision" does this very well at all).
Harami GM, Gyimesi M, Kovacs M. 2013. From keys to bulldozers: expanding roles for winged helix domains in nucleic-acid-binding proteins. Trends Biochem Sci 38(7):364-371
Howard-Flanders P, Boyce RP. 1966. DNA Repair and Genetic Recombination: Studies on Mutants of Escherichia coli Defective in These Processes. Radiat Res Supp 6:156-184
Lee Theriot's X-ray sensitive mutant https://thebradentontimes.com/lee-frances-theriot-p24426-154.htm I was a postdoc with Joseph Greenberg working in the same field at the time
This is well documented in Advances in Microbial Physiology, Volume 16, Moseley and Williams, pages 116-117 (searchable via Google Books).
Mattern et al. (1966) designated mutations that made E. coli Bs-1 sensitive to X-ray radiation as exr (X-ray sensitivity) mutations. That same year, Howard-Flanders and Boyce (1966) independently designated equivalent mutations in E. coli K-12 as lex (locus for X-ray sensitivity). In both strains these mutations were close to the malB locus (b4039; lexA is b4043). A year later, Greenberg was using a broader designation with the same causal root (Loci for Radiation Sensitivity) to designate these same mutations.