I'm puzzled as to where Graham's suggestion about the morphology could have come from. The final -a is just the normal short a ("schwa") at the end of the dictionary form of masculine and neuter nouns.
The word is derived, as I understand it, from the stem श्रम्
śram, which has forms meaning meaning "hard work", "exertion", "act of austerity", "spiritual exercise", so an
āśrama is a hermitage or retreat, a place for spiritual devotion, particularly through austerity. (see
http://www.sanskrit-lexicon.uni-koeln.de/cgi-bin/monier/serv... , bottom of 1st column: see also श्रमण
śramaṇa, middle of 2nd column.)