Rough and ready, but some of the most common unusual terms:
Fern-owl: not quite an actual owl, though in the same family. More commonly now know as a nightjar. The Royal Society for the Protection of Bird’s page has sounds and video.
Hirundines (or hirundinae, hirundinidae): swallows, martins, and swifts (swifts have since been reclassified as not being related to this family). White’s particular favorites; many passages of the diaries concern the then-vexed question of whether these little birds migrate or hibernate when they disappear every year.
Lodged: as in wheat or hay. ‘Lodged’ means flattened by wind or heavy rains, hence a pain to reap.
St. Foin (or sainfoin): a kind of hay, or to be more accurate, a forage crop. White’s particular note of the state of his sainfoin is due to the fact that he began to experiment with this crop in 1768, the year he began the journal (according to “A Selborne Year”, 1986, ed. E. Dadswell). Apparently it’s making a comeback.
I’ll be adding more terms as it occurs to me; please use the contact form to flag words that should be included here!