Wikipedia supplies what it calls a 'non-exhaustive' list of possible candidates, some of whom are plainly sillier and less likely than others.
William Herbert (the Earl of Pembroke). Herbert is seen by many as the most likely candidate, since he was also the dedicatee of the First Folio of Shakespeare's works. However the "obsequious" Thorpe would be unlikely to have addressed a lord as "Mr".You'd think, given this wearyingly long list of possibles, that Shakespearians had exhausted all the relevant contemporaries whose initials were 'W.H.'. But not so! Behold my spanking new theory: the dedicatee is William Howard.
Henry Wriothesley (the Earl of Southampton). Many have argued that 'W.H.' is Southampton's initials reversed, and that he is a likely candidate as he was the dedicatee of Shakespeare's poems Venus & Adonis and The Rape of Lucrece. Southampton was also known for his good looks, and has often been argued to be the Fair Youth of the sonnets; however, the same reservations about "Mr." also apply here.
A simple printing error for Shakespeare's initials, 'W.S.' or 'W. Sh'. This was suggested by Bertrand Russell in his memoirs, and also by Foster and by Jonathan Bate. Bate supports his point by reading 'onlie' as something like 'peerless', 'singular' and 'begetter' as 'maker', ie. 'writer'. Foster takes "onlie" to mean only one, which he argues eliminates any particular subject of the poems, since they are addressed to more than one person. The phrase 'Our Ever-Living Poet', according to Foster, refers to God, not Shakespeare. 'Poet' comes from the Greek 'poetes' which means 'maker', a fact remarked upon in various contemporary texts; also, in Elizabethan English the word 'maker' was used to mean 'poet'. These researchers believe the phrase 'our ever-living poet' might easily have been taken to mean 'our immortal maker' (God). The 'eternity' promised us by our immortal maker would then be the eternal life that is promised us by God, and the dedication would conform with the standard formula of the time, according to which one person wished another "happiness [in this life] and eternal bliss [in heaven]". Shakespeare himself, on this reading, is 'Mr. W. [S]H.' the 'onlie begetter', i.e., the sole author, of the sonnets, and the dedication is advertising the authenticity of the poems.
William Hall, a printer who had worked with Thorpe on other publications. According to this theory, the dedication is simply Thorpe's tribute to his colleague and has nothing to do with Shakespeare. This theory, originated by Sir Sidney Lee in his A Life of William Shakespeare (1898), was continued by Bernard Rowland Ward in his The Mystery of Mr. W.H. (1923), and has been endorsed recently by Brian Vickers, who notes Thorpe uses such 'visual puns' elsewhere. Supporters of this theory point out that "ALL" following "MR. W. H." spells "MR. W. HALL" with the deletion of a period. Using his initials W.H., Hall had edited a collection of the poems of Robert Southwell that was printed by George Eld, the same printer for the 1609 Sonnets. There is also documentary evidence of one William Hall of Hackney who signed himself 'WH' three years earlier, but it is uncertain if this was the printer.
Sir William Harvey, Southampton's stepfather. This theory assumes that the Fair Youth and Mr. W.H. are separate people, and that Southampton is the Fair Youth. Harvey would be the "begetter" of the sonnets in the sense that it would be he who provided them to the publisher, after the death of Southampton's mother removed an obstacle to publication. The reservations about the use of "Mr." do not apply in the case of a knight.
William Himself (i.e., Shakespeare). This theory was proposed by the German scholar D. Barnstorff, but has found no support.
William Haughton, a contemporary dramatist.
William Hart, Shakespeare's nephew and male heir. Proposed by Richard Farmer, but Hart was nine years of age at the time of publication, and this suggestion is regarded as unlikely.
William Hatcliffe of Lincolnshire, proposed by Leslie Hotson in 1964.
Who He. In his 2002 Oxford Shakespeare edition of the sonnets, Colin Burrow argues that the dedication is deliberately mysterious and ambiguous, possibly standing for "Who He", a conceit also used in a contemporary pamphlet. He suggests that it might have been created by Thorpe simply to encourage speculation and discussion (and hence, sales of the text).
Willie Hughes. The 18th-century scholar Thomas Tyrwhitt first proposed the theory that Mr. W.H. and the Fair Youth were one "William Hughes," based on presumed puns on the name in the sonnets. The argument was repeated in Edmund Malone's 1790 edition of the sonnets. The most famous exposition of the theory is in Oscar Wilde's short story "The Portrait of Mr. W. H.," in which Wilde, or rather the story's narrator, describes the puns on "will" and "hues" in the sonnets, (notably Sonnet 20 among others), and argues that they were written to a seductive young actor named Willie Hughes who played female roles in Shakespeare's plays. There is no evidence for the existence of any such person. However, several scholars in the early 20th century identified other persons with that name as possible candidates.
Who? Well, the same person to whom John Norden dedicated the Vicissitudo rerum in 1600:
And which Charles Howard, Earl of Nottingham, Lord High Admiral of England is that, you ask? Why, this one. The same one who sponsored the Admiral's Men. That theatrical company second only to the Chamberlain's men (Shakespeare's company) in London for The Drama. The same Charles Howard who opposed the closing of the theatres in 1584, thereby earning the gratitude of all professional players (like Shakespeare). Whose company who is linked to the first performances of Shakespeare’s Richard III (indeed, latest scholarly opinion is that, at some point early in his career, Shakespeare must have been attached to the Admiral's Men himself). That Charles Howard. Might Shakespeare have good reason to dedicate his sonnets to the Earl's son, William Howard, born right at the end of 1577 and accordingly a toothsome 20-21 in 1598-99 when the sonnets were probably written? Who must have looked something like this:To the Right honora-ble Sir VVilliam Howard knight, theLord Howard of Effingham, Sonne and heyreapparent to the Right Honorable Earle ofNottingham, Lord high Admirall of England.
It's all speculation, of course; but likelier than some. That Shakespeare, writing sonnets perhaps during the plague-related temporary closure of the theatres, might dedicate them to the son of the man who had kept the theatres open a decade and some years earlier? That he would have known the son of the patron of the company where he had worked, as a young man? That, talking not to the Earl himself (Lord High Admiral of England is pretty high), but to his son, a 'Mr' might not seem too demeaning? (William was a knight, but to re-quote the W. H.-ikipedia from above, 'The reservations about the use of "Mr." do not apply in the case of a knight') NOT TO MENTION!--lots of 'how' and 'ward' punning in the sonnets. To take the latter; and starting with one of the procreation sonnets ('The first 17 poems, traditionally called the procreation sonnets, are addressed to a young man urging him to marry and have children in order to immortalize his beauty by passing it to the next generation'):
Neither in in-ward worth nor out-ward fair,Shakespeare was fond of this:
Can make you live your self in eyes of men. 
As thus; mine eye's due is thy outward part,He talks of 'sure wards of trust!' ; splits sonnet 50 between 'how' and 'ward' (first line; 'How heavy do I journey ...'; last line: '... My grief lies onward); tells the addressee that he has 'prison[ed] my heart in thy steel bosom's ward' ; plays cryptic-clue games ('Or as the wardrobe which the robe doth hide'; take away the robe from wardrobe and you have the last syllable of W.H.'s name). Sonnet 51 ends with 'will' and '[ho]ward' in close proximity:
And my heart's right, thy inward love of heart. 
'Since from thee going, he went wilful-slow,Then there's sonnet 62, which takes on a new meaning in the light of this hypothesis:
Towards thee I'll run, and give him leave to go.'
Sin of self-love possesseth all mine eye'Self love' because he (Shakespeare) is called Will, and he is in love with a man called Will; and that fourth line sits up a begs to be read as 'It is so grounded Howard in my heart'. Of 'How's, short for 'Howard' there are many many more. 'Who taught thee, How[ard], to make me love thee more?' 150; better still, 120:
And all my soul, and all my every part;
And for this sin there is no remedy,
It is so grounded inward in my heart.
O! that our night of woe might have remember'dSee? My deepest sense, Howard true sorrow hits! Really, once you start looking you find yourself thinking: how did nobody spot this before?
My deepest sense, how hard true sorrow hits,
And I haven't even started on on my 'buried nautical references, by way of punningly honouring the son of the man who saved England from the Spanish Armada' theory yet.