What personality traits affect ability to cope with pain? It has been shown that pain impairs cognitive function,  in this question and the following studies on the effects of  headaches on cognition. Moor et al write


  It must therefore be emphasised that headache pain appears to impair
  general task performance, irrespective of task complexity, rather than
  specific attentional mechanisms. Headache pain has an effect on the
  core cognitive components necessary for the successful completion of
  tasks, and in particular those involving the updating of the cognitive
  system.


Eccleston et al have shown that the  degree to which pain disrupts  cognitive processes  is  related to facts regarding the pain and the individuals psycho-social activities. 


  The interruptive function of pain depends on the relationship between
  pain-related characteristics (e.g., the threat value of pain) and the
  characteristics of the environmental demands (e.g., emotional
  arousal).


So, it seems that there is physiological and psychological factors that contribute to why pain interferes with cognition.        

Are there personality traits that correlate with an individual's ability to cope with pain?

References


Headache impairs attentional performance.  Moore DJ, Keogh E, Eccleston C.  Centre for Pain Research, The University of Bath, Bath BA2 7AY, UK.
Pain demands attention: A cognitive–affective model of the interruptive function of pain.  Eccleston, Chris; Crombez, Geert   Psychological Bulletin, Vol 125(3), May 1999, 356-366. doi: 10.1037/0033-2909.125.3.356
Gathered from M. Pecina et al., “Personality Trait Predictors of Placebo Analgesia and Neurobiological Correlates,” Neuropsychopharmacology, doi: 10.1038/NPP.2012.227, 2012:


  A recent study has found that personality traits appear to influence
  how strongly a person responds to a placebo treatment for pain.


http://www.psychologytoday.com/blog/unique-everybody-else/201301/personality-and-pain-relief


  Pain relief tended to be stronger in participants who rated themselves
  higher in the personality traits of ego resiliency and agreeableness
  and lower in neuroticism.
  [...]
  The facets that predicted placebo response most strongly were high altruism and straightforwardness (facets of agreeableness) and low angry hostility (a neuroticism facet that is also related to low agreeableness).


More thorough explanations are found in the link provided.
Further information on the same study:
http://www.the-scientist.com/?articles.view/articleNo/33300/title/Personality-Predicts-Placebo-Effect/