The website Wikipedia (whose entries should be taken cum grano salis unless corroborated elsewhere) gives numerous pseudonyms said to have been used by H.P.L. This page is an effort to untangle their origins, at least two of which are not mere inventions.
i) Lewis Theobald
This is an allusion to the historical Lewis Theobald (1688-1744), playwright, man of letters and compiler of the first variorum edition of the works of William Shakespeare (Shakespeare Restored: or a Specimen of the many Errors as well Committed as Unamended by Mr Pope in his late edition of this poet; designed not only to correct the said Edition, but to restore the true Reading of Shakespeare in all the Editions ever published.) Pope's revenge was to make Theobald the first of his Dunces, which has unfortunately ever after tarnished his considerable scholarship, although it was not combined with altogether blameless conduct.
ii) Humphrey Littlewit, Esq.
This name applies at once to a pseudonym and character, the 228-year-old High Tory author-narrator (born precisely two hundred years before H.P.L.) of the 'A Reminiscence of Dr. Samuel Johnson' in which Lovecraft parodies his own beautifully antique diction. Note that the initials agree with Howard Lovecraft Littlewit himself is mauled badly in his first conversation with Dr. Johnson, which I am sure Lovecraft took great pleasure in writing!
iii) Ward Phillips
Phillips was the maiden-name of Lovecraft's mother, as also of his beloved grandfather Whipple Van Buren Phillips. It is possible that ''Ward'' is a mere contraction of Howard. It is a distinct element in the name, which comes from Old Norse Hávarðr denoting high-guard.
iv) Edward Softly
This appears to be a complete invention, though the second element of ''Edward'', as ''Howard'', denotes ''guard'' in a primitive Germanic language, Eadweard , guard-of-riches, cf. the verbal form ''to ward''.