Why did the process of sleep evolve in many animals? What is its evolutionary advantage?
This good non-scholarly article covers some of the usual advantages (rest/regeneration).

One of the research papers they mentioned (they linked to press release) was Conservation of Sleep: Insights from Non-Mammalian Model Systems by John E. Zimmerman, Ph.D.; Trends Neurosci. 2008 July; 31(7): 371–376. Published online 2008 June 5. doi:  10.1016/j.tins.2008.05.001; NIHMSID: NIHMS230885. To quote from the press release:


  Because the time of lethargus coincides with a time in the round worms’ life cycle when synaptic changes occur in the nervous system, they propose that sleep is a state required for nervous system plasticity. In other words, in order for the nervous system to grow and change, there must be down time of active behavior. Other researchers at Penn have shown that, in mammals, synaptic changes occur during sleep and that deprivation of sleep results in a disruption of these synaptic changes.