From stock + dove, because once believed to be the stock of the domestic pigeon, or perhaps from its breeding in the stocks (trunks) of trees.
stockdove (plural stockdoves)
Part or all of this entry has been imported from the 1913 edition of Webster’s Dictionary, which is now free of copyright and hence in the public domain. The imported definitions may be significantly out of date, and any more recent senses may be completely missing.
(See the entry for “stockdove”, in Webster’s Revised Unabridged Dictionary, Springfield, Mass.: G. & C. Merriam, 1913, →OCLC.)