Only NaiveBayes in WEKA currently makes use of attribute weights. However, you can rescale the attributes according to their importance and this will affect the result of distance-based clustering algorithms. Just make sure you turn normalisation off when you do this, so the attributes are not put on the same scale again! There is a parameter in the distance functions in WEKA that allows you to turn normalisation off.
You can use the MathExpression filter to scale an attribute.
The exact scaling function to apply depends on the distance function. It’s most straightforward in the case of Manhattan distance: if you have three attributes with weights 0.5, 0.3, and 0.2, just multiply each of the attributes’ values by the corresponding weight. In the case of the Euclidean distance, you could use the square root of the weight as the multiplier.