This recipe for dill pickles is from 1638 - too late.
However, there is this one from 1609:
You may take a gallon of faire water, and a pottle of veriuyce, and a pinte of
bay salt, and a handfull of greene Fennell or Dill: boile it a little, and
when it is cold put it into a barrell, and then put your Cowcumbers into that
pickle, and you shall keepe them all the yeere."
(notes - A pottle is 4 pints. Bay salt is vague, I used sea salt)
Which could point to it being even earlier.
This recipe is not specifically for a cucumber but for any veggies - and I've seen plenty of evidence for "cowcumbers" in the 16th century so this might be it!
Flavor caraway seeds and anise with pepper and with vinegar and with honey. And make it gold with saffron. And add thereto mustard. In this condiment you may make sulze (pickled or marinated) parsley, and small preserved fruit and vegetables, or beets, which(ever) you want. (Guoter Spise)
Not exactly a dill pickle but at least it's a start. However, the florilegium files show that most people in the SCA seem to think or know that there is an Elizabethan recipe out there for dill pickles - which works with what the NY Food Museum says. So, dill pickles it is!