It's St David's Day today, so I looked up the word daffodil in the OED. And a very interesting history it has too. The Dictionary says it does not know how the initial d crept in to the word - the flower was originally affodill, belonging to the Asphodelus genus. Suggestions as to where the d came from, according to the Dictionary, include: that it followed the same process that saw Edward become Ted; that the d is based on the th of 'the daffodil', or northern English t'daffodil; that the d came from the word and, as in 'fennell an-d affodil'; that it came from the Dutch or Flemish article, de affodil; or from the French de meaning 'of', as in fleur d'aphrodille.
An affodill was originally the flower asphodel, but the name then mistakenly came to be applied to the narcissus. Botanists retained the name affodil for the asphodel (a flower with spiky petals, see here), and applied the name daffodil to the yellow narcissus.
Other words for the daffodil in the OED are: bell-rose, chalice-flower, churn, yellow crow-bells, daffadowndilly (or daffydowndilly), hen and chickens, hoop-petticoat, and Lent lily.
The Welsh for daffodils is Cennin Pedr, which translates as (St) Peter's leeks. The Welsh word for leeks is cennin, which explains why both daffodils and leeks are Welsh symbols. Here's a BBC page on more Welsh words and traditions.