There are many varieties of speedwell, and I'm pretty sure that the one I've got is Germander Speedwell, or Veronica chamaedris by its latin name. Most information on its use seems to draw on Mrs Grieve's Modern Herbal which says,
The Germander Speedwell has a certain amount of astringency, and an infusion of its leaves was at one time famous for coughs, the juice of the fresh plant also, boiled into a syrup with honey, was used for asthma and catarrh.That's what I need!
I picked a handful of shoots and added boiling water to make a tea. I didn't like the taste too much on its own, but with a little honey it was quite nice (I often drink mint tea with honey, too). As for the catarrh, it's hard to be sure, but this seems to be working a lot better than the commercial cough medicine I bought yesterday.
* I didn't manage to find a definitive answer to this question. Advice on eradicating it from lawns (why??) included letting the grass grow long to act as a living mulch and shade out the speedwell. On the other hand, it's also described as an invasive weed that's hard to get rid of, so I suspect it will survive most things. I'll try not to bury it too deeply, in any case. And in case you were wondering, this is related to maintenance of the terrace, which I'll write more about shortly.