Typically, these "50% genetic" claims are misleading simplifications of the longer claim: "50% of the variance in the population can be explained by genetic differences." It is a measure of how accurate a statistical model could be, if it only considered one factor. That appear to be how it is intended here.
The 50% figure seems to have been popularised in a book The How of Happiness by Dr Sonja Yubomirksy, a professor at the University of California.
This conclusion was drawn in her 2005 paper, Pursuing Happiness: The Architecture of Sustainable Change (with co-authors, Kennon M. Sheldon and David Schkade):
First is the idea of a
genetically determined set point (or set range)
for happiness. Lykken and Tellegen (1996)
have provided evidence, based on twin studies
and adoption studies, that the heritability of
well-being may be as high as 80% (although a
more widely accepted figure is 50%; Braungart,
Plomin, DeFries, & Fulker, 1992; Tellegen et
al., 1988; cf. Diener et al., 1999). Whatever the
exact coefficient, its large magnitude suggests
that for each person there is indeed a chronic or
characteristic level of happiness
The OP expressed skepticism about the quality of Happiness research, so let's look at those references:
Genetic influence on tester-rated infant temperament as assessed by Bayley's Infant Behavior Record: Nonadoptive and adoptive siblings and twins.
By Braungart, Julia M.; Plomin, Robert; DeFries, J. C.; Fulker, David W.
Developmental Psychology, Vol 28(1), Jan 1992, 40-47.
This looked at the temperament (along three factors: affect-extraversion, activity, and task orientation) for infant twins. "Both the sibling adoption and twin data yielded evidence for significant genetic influence for the 3 IBR factors at 12 and 24 mo: Heritability estimates ranged from 35–57%."
So, this didn't measure adults and didn't measure "happiness" but external temperament, but suggests that personality is heritable.
Tellegen, A., Lykken, D. T., Bouchard, T. J., Wilcox,
K. J., Segal, N. L., & Rich, S. (1988). Personality
similarity in twins reared apart and together. Journal
of Personality and Social Psychology, 54,
A personality questionnaire, MPQ was administered to over 300 pairs of adult twins.
Some of the factors that were measured include "Well-Being" and "Positive Emotionality".
The variance of "Well-Being" was 48±8% accounted for by genetics.
The variance of "Positive Emotionality" was 40±8% accounted for by genetics. [See Table 5]
Lykken and Tellegen (1996), Happiness is a stochastic phenomenon:
This paper builds on the results of the previous paper.
Happiness, or subjective well-being, was measured
on a birth-record-based sample of several thousand middle-aged
twins using the Well-Being (WB) scale of the Multidimensional
Personality Questionnaire. Neither socioeconomic status,
educational attainment, family income, marital status, nor
an indicant of religious commitment could account for more
than about 3% of the variance in WB. From 44Vc to 52% of the
variance in WB, however, is associated with genetic variation.
Based on the retest of smaller samples of twins after intervals of
4,5 and 10 years, we estimate that the heritability of the stable
component of subjective well-being approaches 80%.
This 80% figure seems to be partly speculative estimate based on the expected long-term retest stability of the questionnaire. (That is, how much someone will give the same sort of answers to the questionnaire ten years later.)
Diener, E., Suh, E. M., Lucas, R. E., & Smith, H. L.
(1999). Subjective well-being: Three decades of
progress. Psychological Bulletin, 125, 276–302.
This is a review paper, and examines many of the results above, and compares them to other results, suggesting caution with the 80% figure quoted above.
I can't see anywhere in the paper where they draw a firm figure - the paper is more qualitative, than quantitative - but I would suggest if you wanted to understand the general state of play of the research (as of 1999), this would be a good paper to read.
There are several studies that measure factors of human emotional temperament and well-being that could reasonably described as "happiness" in an informal setting. Genetics account for a large part of the variability (i.e. you can make a good guess about someone's temperament if you subject their identical twin to a questionnaire), but the exact percentage of the variability accounted for varies between studies techniques. A 50% estimate is a reasonable short-hand.