what does "nurse her elbow"mean?
On the couch opposite the stateroom desk Bird fingered her choker of pearls and jotted down her memoirs. They formed a jumbled chromo. She overheard "the agonized voice" of a Secret Service man crying that the Service had never lost a President before, and she "hurt for him." She was aware of Lyndon, moving restlessly from chair to couch to chair. She intercepted a message from Parkland—Mrs. Kennedy wouldn't move from Major Medicine without the body. Johnson let it be known that 26000 wouldn't budge; they were going to "wait on" the gallant lady and the coffin. Television's square eye had become somewhat newsier; there were reports that the President had been the victim of a .30-.30 rifle. The stateroom's occupants kept loosening their collars and mopping their foreheads. Lady Bird alone felt chilled. She nursed her elbows and listened to Lyndon canvassing the men on the question of where he should be sworn in.

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It means that she "tended" to her elbows. I do not have enough information, but we usually use this verb (to "nurse") in reference to a wound or at least pain or discomfort in a part of our bodies, so I am guessing that something was wrong with Mrs. Johnson's elbow. That is why she nursed her elbow.

(PS: "to nurse" also has another meaning when used in the context of a mother and her infant, but this is obviously not the case here)

Answer

It means that her elbows were previously injured somehow and she is now paying attention to them. She could be rubbing her elbows to make them better or something along these lines.  A definition would be to treat carefully, especially in order to prevent pain.

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