The original paper from Lederberg from January 1953 (see reference 1) indeed doesn't mention the origin of the name, but the paper in reference 2 does. It says:
The isolation of λ was first reported in 1951 by Esther Lederberg
(119), then a Ph.D. student at the University of Wisconsin, and later
was described, in greater detail, in a 1953 Genetics paper by Esther
and Joshua Lederberg (120). The discovery was accidental, when a
λ-sensitive strain of Escherichia coli K-12 (W518, obtained after UV
irradiation) was crossed with its parent. The mixture yielded plaques,
and the source of the virus was the K-12 parent. W518 cells that
survived infection became stable lysogens which, like the K-12 parent,
were immune to superinfection and which released unaltered phage.
Although the Lederbergs were initially opposed to the notion, crosses
between lysogens and sensitive cells led them to suggest that λ
prophage was chromosomal and linked to gal. Joshua Lederberg recalls
that he was convinced, from earlier work of Burnet and Lush (21), that
lysogeny was a real phenomenon but that he “fully expected lambda to
be a [plasmid]—in fact the term lambda was modeled after Sonneborn's
kappa [in Paramecium; see reference 156 for a recent review], so it
was quite a shock to discover the contrary.” (In fact, years later
Hideo Ikeda and Jun-ichi Tomizawa  showed that prophage P1, unlike
λ, is a plasmid and not part of the host chromosome!)
The mentioned kappa particle in Paramecium is an inheritable cytoplasmatic symbiont and the Lederbergs thought of something similar for their discovery. My assumption here is that lambda was the next greek letter, hence the name. The publication in which Sonneborn describes Kappa can be found in reference 3, a short historical overview on the discovery of Kappa in the review in reference 4.
- Genetic Studies of Lysogenicity in Escherichia Coli.
- Little lambda, who made thee?
- Mating Types in Paramecium Aurelia: Diverse Conditions for Mating in
Different Stocks; Occurrence, Number and Interrelations of the Types
- R-body-producing bacteria.