The majority of the constellations were named after various creatures that the 16th century explorers had encountered (e.g. Bird of Paradise, Chameleon, Toucan, Flying Fish) and there is no earlier mythology associated with them.
Beta Tucanae is in fact a group of six stars which appear to be at least loosely bound into a system. The two brightest of these, Beta Tucanae A and Beta Tucanae C, are 27 arcseconds apart and have magnitudes of between 4 and 5. They are accompanied by a third star which is further away, separated by 9 arcminutes from them.
Kappa Tucanae is a group of four stars: two binary stars.
Lambda Tucanae is an optical double - that is, the name is give to two stars which appear close together from our viewpoint, but are in fact far apart in space. The two stars are known as Lambda 1 and Lambda 2. Lambda 1 is itself a binary star, with two components.