First - you might want to redefine you search. Are you looking for happiness or rather positive affect? Happiness is fairly ambigious term, and it's much more associated with positive psychology studies on well-being. If you are interested in more global definition of happiness, check the work of Mihaly Csikszentmihalyi.
On the other hand, there is a large number of studies on physiological measurements of positive affect.
One such physiological measurement is Electromyography (EMG) - recording the electrical activity produced by skeletal muscles. EMG will detect very brief smiles or higher activity in cheek muscles (zygomaticus major) which are correlated with positive affect. There is quite classic (but very quoted) paper on that:
Cacioppo JT, Petty RE, Losch ME, Kim HS. (1986) Electromyographic
Activity Over Facial Muscle Regions Can Differentiate the Valence and
Intensity of Affective Reactions. J Pers Soc Psychol., 50(2):260-8. download
Another simple physiological assesment is heart rate measured by the interbeat interval (IBI). For example, study by Brosschot & Thayer (2003) shows that heart rate response is longer after negative emotions than after positive emotions.
Brosschot JF, Thayer JF. (2003) Heart rate response is longer after
negative emotions than after positive emotions. Int J Psychophysiol.,
In fact, the full spectrum of somatic measurements have been used along heart rate including pulse transmission time to the finger, skin conductance level or pupil dilation (Partala, 2003). All those are a bit less reliable methods and usually they detect arousal rather then indicate physiological differences between positive and negative affect.
Partala T.; Surakka V. (2003) Pupil size variation as an indication of
affective processing. International Journal of Human-Computer Studies,
Finally, I would advise browsing literature on measurements of negative affect. You are likely to find some interesting methods there, like in this paper on the psychophysiology of crying (Gross et al., 1994).
Gross JJ, Frederickson BL, Levenson RW. (1994) The psychophysiology of
crying. Psychophysiology, 31(5):460-8. download