There appears to be a higher correlation with environmental influences affecting handedness, as opposed to the genetic factors; although there is conclusive genetic evidence, it is not the major contributing factor in determining handedness.
I quote from the link @what provided in the comments.
Handedness displays a complex inheritance pattern. For example, if
both parents of a child are left-handed, there is a 26% chance of that
child also being left-handed. A large study of twins from 25,732
families by Medland et al. (2006) has indicated that the heritability
of handedness is roughly 24%. This leaves about three quarters of
the effect to be explained by environmental factors.
To date, two theoretical single gene models have been proposed to
explain the patterns of inheritance of handedness, the first by Dr.
Marian Annett of the University of Leicester and the second by
Professor Chris McManus of UCL.
Both models propose that there is a variant in a single gene that has
two alleles. Carriers of one allele are more likely to be
right-handed, and the other allele does not specify the direction of
handedness, instead leaving it to chance. They differ on the precise
effect of the 'right-shift' allele, but both models provide similar
fits to data on the inheritance of handedness. Oxford University
psychiatrist Professor Tim Crow has taken the single-gene model one
step further, and proposed that mutations in the gene PCDH11X were
responsible for the evolution of handedness, cerebral asymmetry,
language, susceptibility to schizophrenia, and was the speciation
event that created Homo sapiens.
Although single-gene models can be fitted to the data, a number of
genetic linkage studies have been performed, all of which have
provided evidence for different regions of the genome contributing to
variation in handedness.  Only one of these studies
has led to the identification of a specific gene that is proposed to
contribute to variation in handedness. 
Medland et al. found a gene that is positively correlated with
left-handedness in females, and negatively correlated in males. This
may help to explain why there are more left-handed men than women
(around 12% in men versus 10% in women globally).3
It is, also, important to define handedness, or types of handedness, and the variability between self-reporting and objective evidence.
However, genes influencing sidedness remain elusive. We measured
direction and consistency of hand, foot, and eye preference in 584
Mexican-Americans from families participating in the San Antonio
Family Diabetes/Gallbladder Study. Using maximum-likelihood-based
variance components methods, we estimated weak (.11 ≤ h2≤.17) but
significant heritability for foot preference, eye preference, several
hand preferences (writing, drawing, throwing, using scissors, using
spoon, striking match), and a composite hand preference trait.
Self-reported handedness was significantly heritable (h2=.57), whereas
hand preference for opening a box or using a toothbrush or knife was
not. Many trait pairs had significant genetic correlations, and all
had significant environmental correlations.
(Heritability and linkage analysis of hand, foot, and eye preference in Mexican Americans Diane M. Warrena*, Michael Sternb, Ravindranath Duggiralac, Thomas D. Dyerc & Laura Almasyc DOI:10.1080/13576500600761056)
There is also the concept that handedness is on a continuum and that there is not, necessarily, a definitive left or right handedness, bit rather where the individual sits on the contiuum of cerebral dominance. This study also examines the genetic modelling versus the environmental factors, and the unreliability of self-reporting. It also, introduces the idea the people may under-report left handedness. This also demonstrates that left handedness is inherited maternally.
Whereas many assume there is a true incidence of left-handedness, the
RS theory suggests that degrees of hand preference map onto a
continuous baseline of asymmetry that can be cut at any point to
represent observed incidences of left-handedness. The parameters of
the RS genetic model were derived from findings for speech
lateralization in aphasics, supporting the argument that the relevant
genetic locus is “for” cerebral dominance, not handedness. Genetic
predictions are given for two levels of parental left-handedness (10%
and 20%). Studies of family handedness distinguishing for sex in both
generations gave generally good fits where handedness was assessed by
self-report, but more variable fits for indirect report of relatives'
handedness. The tendency to find a higher proportion of left-handed
children born to left-handed mothers than left-handed fathers is not
likely due to X-linked inheritance, but rather to slightly stronger
expression of the RS+ gene in females than males, and also
under-reporting of left-handedness in mothers by right-handed
(The genetic basis of lateralization Marian Annett dx.doi.org/10.1017/CBO9780511576744.006)
There has been research demonstrating that the LLRTM1 gene contributes to handedness. This gene has also been linked with increased incidence of schizophrenia. An interesting observation, given the historic "demonisation" of left handers. Could this be, because there was a perceivable increased incidence of "insanity" amongst the left handers? (note, my own theory)
Left-right asymmetrical brain function underlies much of human
cognition, behavior and emotion. Abnormalities of cerebral asymmetry
are associated with schizophrenia and other neuropsychiatric
disorders..... This is the first potential genetic influence on
human handedness to be identified, and the first putative genetic
effect on variability in human brain asymmetry. LRRTM1 is a candidate
gene for involvement in several common neurodevelopmental disorders,
and may have played a role in human cognitive and behavioral
LRRTM1 on chromosome 2p12 is a maternally suppressed gene that is associated paternally with handedness and schizophrenia. Francks C, et al.
So, given my example, it is not surprising that:
- The left handed family members were all on the maternal side.
- The incidence of left handedness was low.