Luna was diagnosed with early onset schizoaffective disorder, depressive type, after her mother passed away and she had her first psychotic episode, and then a little after fell into a depressive spell.
 
Many of her ‘odd’ behaviors are actually magical treatments for hallucinations, delusions and disorganized thinking, as well as little tricks to lessen the symptoms of her depression. 

For example, dirigible plums help lessen the effects of disorganized thinking, and wearing them in her ears helps even more than eating them, and her butterbeer cork necklace is a lucky charm to help reduce anxiety around hallucinations and delusions. 

She also keeps her wand tucked behind her ear because of a belief that she could die from a backfiring curse, like her mother, without her holding it, thus keeping it tucked where it can’t hit her, unlike most the wizarding population, who keep it in their pockets.