I hadn't gotten this reply yet and in the end I tried coriander and parsley, but they were overpowered by my dressing, I did not taste them much.
I agree chervil would make the best substitute, but I didn't have that either - that would have been my first thought.
Might try with basil indeed next time, but the flavour is very different from tarragon! Or the fennel seed or anise seed suggestion.
I would think fennel fronds YES. And as I frequently buy fennels, I am even thinking of mincing and cutting the fronds, as a regular substitute for tarragon or chervil.