"Mary Frances says she named him Paul because she liked the name and Alpheus because at the time, a Temperance lecturer, whose name was Alpheus, was creating a great sensation in St. Louis, so it occurred to her to give that name to her boy."

From an Edwardsville Newspaper:
Dispatches from Havana, Cuba, tell of the death on February 10, 1930, of Alpheus Paul Berry, 82, a former resident of Edwardsville, residing in Cuba since the close of the Spanish-American War. A brother, Hon. Wm. H. Berry, former treasurer of Pennsylvania and for many years a national power in Democratic politics, died two years ago. Mrs. Jenny B. Fulton, a sister residing in Richmond, Va., survives.
Mr. Berry, who gave service in the 30th Illinois Regiment during the Civil War and in the Engineering corps during the war with Spain, gained considerable prominence in Cuba as the only survivor of both wars living on that island. On the occasion of Coolidge's visit to Cuba, Mr. Berry was an honored guest at his reception.
Mr. Berry's only immediate survivor is a son, Edward D. Berry, a lecturer in journalism at Northwestern University.
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When Alpheus was stationed in Cuba, he corresponded with his wife until about 1905. Then he stopped correspondence. Frances tried to find out what happened to him and finally filed for Widow's Pension in 1913. She was denied over & over because there was no proof of his death. That dragged on until 28 May 1923 when he suddenly applied for his own pension. Frances then filed for half of his pension due to desertion but was denied because her husband was not a resident of the US. Frances died 4 months later. Then after Alpheus died, a "common-law" wife tried to apply for the rest of his pension. She was denied.