The problem was also studied by Besicovitch from the geometric measure-theoretic point of view in the 1930s. In particular, he Besicovitch was motivated by the problem of determining the sets of reals on which the variation of any continuous monotone function vanishes. Besicovitch He proved in 1934 that assuming CH there exists an uncountable set of reals which has strong measure zero (see "Concentrated and rarified sets of points", Acta Mathematica, (1934), Vol. 62, pp. 289-300).
Besicovitch actually constructed what he called a concentrated set (an uncountable set of reals $E$ is said to be concentrated on a countable set $H$ iff for any open set $U$ if $H \subset U$, then $E \setminus U$ is countable). The Besicovitch concentrated sets can be viewed as a weaker measure-theoretic analogue of Lusin sets.
The earlier stages of the study theory of strong measure zero sets are summarized in Sierpinski's monograph "Hypothèse du continu" (which has been cited recently in connection with another MO question). Sierpinski refers to these objects as the sets satisfying Property C (see the definition on p. 37).
[EDIT. Somewhat surprisingly the work of Besicovitch and Sierpinski on strong measure zero sets is not mentioned at all in the book recommended by Andres. The historical development of the theory is discussed in detail in the fundamental survey article "History of the Continuum in the 20th Century" by Steprans.]
