What is a word for a place to study and rest?

Meditation

from Old French meditacion "thought, reflection, study", and directly from Latin meditationem (nominative, meditatio), "a thinking over, meditation," from meditatus, pp. of meditari "to meditate, to think over, consider," frequentative form from PIE
base *med- "to measure, limit, consider, advise" (cognates: Greek medesthai
"think about," medon "ruler," Latin modus "measure, manner," [...] mederi "to heal," medicus
"physician," ...)

The verb meditate which Oxford Dictionaries define as:

1. Focus one’s mind for a period of time, in silence or with the aid of chanting, for religious or spiritual purposes or as a method of relaxation
   1.1. (meditate on/upon) Think deeply about (something):
   he went off to meditate on the new idea

Thus to name a home Meditation suggests a place which provides the space and necessary conditions to think things over in a calm reflective manner (i.e. to study something in depth) and at the same time, achieve a state of deep relaxation and inner calmness. In other words, a place where to rest one's body and mind.