The obvious answer is June — specifically the third full week of June, when the National Old Time Fiddlers' Contest takes over the town. If you have any interest in American roots music, this is one of the best events in the country, and the informal jamming that spills out around the official competition is as good as the contest itself. Book accommodation well in advance; the town fills up completely.
The rest of the year, Weiser has a high desert climate: hot dry summers, cold winters, and not much precipitation. Spring (April–May) and fall (September–October) are pleasant for passing through. Winters can bring snow and icy roads, but nothing particularly severe by Idaho standards.