The Ligurian coast has a very mild climate and in the Cinque Terre weather is the mildest of all. Summer days can get very hot but there is seldom too much humidity and cooling breezes provide welcome relief from the heat.
In the Cinque Terre the summers are long, dry and lovely. The only thing that spoils them are the crowds during July and August.
Most rain falls during the October/November and March/April periods; however, even in these months you still get some very lovely days squeezed in between the rainy ones.
Many locals call early spring the secret season because there are so few tourists about and the sunny days have a clarity of light and a brightness about them that is not seen at any other time of year. Please don't visit this time of year though - otherwise the locals will know I'm the one who spilled the beans.
Winters are not too cold at all with minimum temperatures seldom dropping below 40F.
November through March is the only period when you will need a coat and need to wrap up warmly, particularly in the evening.
I would recommend late May, June or early September. All three of these months have great weather and not too many tourists either.
If it is a choice of visiting during one of the "bad" months and not visiting at all then go ahead and visit. If you can though try to avoid the Cinque Terre in July or August, because of the crowds more than the weather.
If you do visit in July or August my advice would be to stay in the far less crowded and pleasant town of Levanto..
Levanto is right on the doorstep of the Cinque Terre and the train from Levanto goes up and down the villages of the Cinque Terre all day.
From the point of view of the weather late October and November are best avoided. Yes, you do get great days between the rainy ones but boy, when it rains during these months it buckets down. Even causing flash floods on occasions.
For all the secrets of the Cinque Terre read my guide here.