Is there a name for thin (3/8") redwood/cedar boards for basketweave?
This is called bender board. Typically made of Redwood for rot resistance. 

It is used as edging for gardening and lawns, forms for concrete walkways and base material for making lattice. 

Available on the US West Coast either direct from Redwood products mills or from Lowe's or Home Depot in thicknesses from 1/4" to 3/8"