because your mom does not move her arm, her blood requirement is somewhat less. frequent massage could help. the other question is: does she have as good a pulse in that arm as in the unaffected one? make sure she did not have a blood clot in her arm that's causing her to have less blood, and hence lower temperature there.
Hi. We have experienced the same. The weak side arm and leg (left in our case also) is very cold in comparison to the right arm and leg. We have found what helps is massaging the arm from finger tips all the way up to the shoulder and then the same procedure with her foot and leg. The warmth returns for a period of time as well as the benefit of reducing puffiness or swelling in those limbs which helps movement. And she says it feels so good and is very thankful.
The "why" has been our question also but not getting any definite answer anywhere. So I will be monitoring the responses your post receives. My opinion is that poor circulation in the affected limbs is the cause and massaging periodically stimulates circulation and helps reduce fluid retention as well.