The Caribbean coast operates on a different weather calendar than the rest of Costa Rica. The driest, sunniest months are typically September and October — the inverse of the Pacific side — with a secondary dry window in February and March. The rest of the year sees serious rain, particularly November through January.
The single best reason to time a visit is Carnaval de Limón in mid-October, a week-long Afro-Caribbean festival of parades, calypso bands, and dancing in the streets that traces its origins to a 1949 celebration of the discovery of the Americas. The city books out; reserve accommodation months in advance. Outside Carnival, midweek visits anywhere in the year are quiet — Limón is not a place that fills up except during the festival and Costa Rican school holidays.