June through September is when Sevanavank makes the most sense as a combined monastery-and-lake trip. July and August are peak season — warm, busy, full resort atmosphere along the shore. If you want to swim, this is the window, though the water stays cold at 2,000 metres.
May and October are quieter and often beautiful. October in particular has good autumn colour on the hills around the lake. The monastery is open year-round and sees few visitors in the off-season.
Winter is possible — the churches are accessible and there is something stark and striking about the monastery in snow — but the road from Sevan town can be icy, most restaurants and facilities along the shore will be shut, and the marshrutka frequency drops. Come prepared or join a tour.
Avoid peak summer weekends (Saturday and Sunday in July–August) if you want the monastery to yourself. Arrive early on those days if you cannot avoid them — the tour buses arrive mid-morning.