The history of both the name "Sobolev space" and the notation (which changed over the years), has been well described by J. Naumann [link to pdf file]. The history of the name is particularly amusing:
These spaces, at least in the particular case $p=2$, were known since the very beginning of this century, to the Italian mathematicians Beppo Levi and Guido Fubini who investigated the Dirichlet minimum principle for elliptic equations. Later on many mathematicians have used these spaces in their work. Some French mathematicians, at the beginning of the fifties, decided to invent a name for such spaces as, very often, French mathematicians like to do. They proposed the name Beppo Levi spaces. Although this name is not very exciting in the Italian language and it sounds because of the name ”Beppo”, somewhat peasant, the outcome in French must be gorgeous since the special French pronunciation of the names makes it to sound very impressive. Unfortunately this choice was deeply disliked by Beppo Levi, who at that time was still alive, and - as many elderly people - was strongly against the modern way of viewing mathematics. In a review of a paper of an Italian mathematician, who, imitating the Frenchman, had written something on ”Beppo Levi spaces”, he practically said that he did not want to leave his name mixed up with this kind of things. Thus the name had to be changed. A good choice was to name the spaces after S.L. Sobolev. Sobolev did not object and the name Sobolev spaces is nowdays universally accepted.
Concerning notation, in the 1950's $BL$ was used (for Beppo-Levi function)space), notably by Nikodym. Sobolev himself originally used $L$ before switching to $W$ in his 1950 book. With the demise of the Beppo-Levi name, $W$ became commonplace.
I agree with Nate that the notation $H$ for $p=2$ is a clear reference to Hilbert.