How was the Huntington's disease gene's location found?
A Google search for "Huntingtons disease gene discovery" yielded this page at the Nature Education Scitable website. The following citations are provided regarding the molecular basis of HD:


  Huntingtin (HTT) was the first disease-associated gene to be
  molecularly mapped to a human chromosome (Gusella et al., 1983)2. Ten
  years later, scientists identified the DNA sequence and determined the
  precise nature of the HD-associated mutation in HTT (MacDonald et al.,
  1993).


Gusella et al. used restriction fragment length polymorphism (RFLP) analysis to locate a RFLP marker linked to Huntington's disease on chromosome 4. From the introduction of their paper:


  We have now identified an anonymous DNA fragment from human chromosome
  4 that detects two different RFLPs in a HindIII digest of human
  genomic DNA. This polymorphic DNA marker shows close genetic linkage
  to the Huntington's disease gene in two separate families, although a
  different haplotype of the marker segregates with the Huntington's
  disease gene in each family. We infer that the Huntington's disease
  locus resides on human chromosome 4.


Further details given in the paper might justify the description given in the question. "Fabolous luck" could refer to the discovery or availability of suitable families and pedigrees for the genetic studies. The "steady detective work" would be the genetic mapping and fieldwork required to establish the link. The following quotes describe some of the efforts spent on the project:


  The success of any genetic linkage project depends in large part on
  the quality of family material available for study. We initially
  invested considerable effort in identifying a useful family from the
  National Research Roster for Huntington's Disease Patients and
  Families at Indiana University. An American family of reasonable size
  was selected and blood samples were obtained (...). Subsequently, a
  substantially larger Huntington's disease family was located. This
  pedigree stem from a unique community of interrelated Huntington's
  disease gene carriers living along the shores of Lake Maracaibo,
  Venezuela. (...)For the past three years an expedition has spent one
  month annually in Venezuela collecting pedigree information, tissue
  samples, and clinical data. (...) In Venezuela, many family members
  were examined for three consectuvie years. Permanent lymphoblastoid cell lines were 
  again established for each individual to act as a permanent source of
  genomic DNA. In both families each individual was examined by at least
  one neurologist experienced and knowledgable about Huntington's
  disease.


The Scitable page also summarizes the approach and results described in the paper. For the RFLP analysis, several DNA probes were tried, of which one, "G8", resulted in a HD-linked pattern:


  James F. Gusella and colleagues carried out a study to determine
  whether they could identify a DNA probe that would show an
  HD-associated restriction fragment length polymorphism (RFLP) when
  used in Southern blot analyses of chromosomal DNA digested with the
  restriction enzyme HindIII (palindromic recognition sequence
  5'-AAGCTT-3'). The team identified one probe out of 12 tested, called
  G8, that showed a specific RFLP pattern associated with HD in two
  large families with a history of the disease (Gusella et al., 1983).
  Using the G8 probe, they next identified two HindIII sites (called H1
  and H2) that were palindromic within this chromosomal region. DNA
  fragments at these sites vary in length among different HD lineages.
  Because researchers used two large pedigrees in these experiments,
  they were able to obtain statistical support for their discovery
  (Figure 3).


However, at the time the origin of the G8 probe DNA in the human genome was not known, so human-mouse somatic cell hybrids were made to identify where the G8 probe DNA belonged. This showed that the G8 probe corresponded to chromosome 4, and indicated that the HD gene would be found there:


  Researchers found that the G8 DNA probe hybridized only to Southern
  blots using chromosomal DNA from human-mouse somatic cell hybrid lines
  that contained human chromosome 4. Therefore, researchers determined
  that the G8 DNA probe is located on human chromosome 4, and they
  concluded that the HD gene is located on chromosome 4 near the region
  corresponding to the G8 DNA probe (Figure 5; Gusella et al., 1983).