With regard to the origin of the phrase, Nigel Rees, A Word in Your Shell-like (2004) dates it to no later than 1903:
(the) man on the Clapham omnibus I.e. the ordinary or average person, the MAN IN THE STREET, particularly when his/her point of view i instanced by the courts, newspaper editorials, etc. This person was first evoked (according to a 1903 law report) by Lord Bowen when hearing a case of negligence; 'We must ask ourselves what the man on the Clapham omnibus would think.' Quite why he singled out that particular route we shall never know. It sounds suitably prosaic, of course,and the present 77A to Clapham Junction (1995) does pass through Whitehall an Westminster, thus providing a link between governors and governed. There is evidence to suggest that the 'Clapham omnibus' in itself had already become a figure of speech by the mid-19th century. In 1857, there was talk the 'occupant of the knife-board of a Clapham omnibus'.
John Ayto, Oxford Dictionary of English Idioms (2009) concurs with Rees:
the man on the Clapham omnibus the average man, especially with regard to his opinions. British
This expression is attributed to the English Judge Lord Bowen (1835–94) who used it as a metaphor for any ordinary reasonable person—such as a juror is expected to be. Clapham is a district in South London.
Further detail appears in John Ayto, 20th Century Words (2002) [combined snippets]:
Clapham n man on the Clapham omnibus (1903) the ordinary or average man; the 'man in the street'. British; an evocation of the extreme ordinariness (and hence representativeness) of a commuter travelling in by bus to work in central London from Clapham (in 1903 a south-westerm suburb, by the end of the century, virtually inner-city). Originally largely a legal figment appealed to by lawyers, but latterly in wider use. The general idea seems to have been around as long ago as the mid 19th century: 'So thoroughly has the tedious traffic of the streets become ground into the true Londoner's nature, that ... your dog-collar'd occupant of the knife-board of a Clapham omnibus, will stick on London bridge for half-an-hour with scarcely a murmer', Journal of the Society of Arts (1857)
1903 Law Report (King's Bench Division): 'Fair', therefore, in the collocation certainly does not mean that which the ordinary reasonable man, 'the man on the Clapham omnibus', as Lord Bowen phrased it the juryman common or special, would think correct appreciation of the work.
Another interesting comment on the phrase appears in National Academy of Arbitrators, Proceedings of the ... Annual Meeting (2006) [combined snippets]:
The second quibble is about the word "reasonable." You all heard Dick's description of the evolution of the word in our field. I'm going back just a little further. We all know about what used to be called the "reasonable man." But what he was called by the English judges long before—since the 19th century—was “the man on the Clapham omnibus”—an officious little objective figure with no personal views other than to say, “I disagree.”
The earliest match in Google Books that I've been able to find for "Clapham omnibus" is from Thomas Hood, "The Confessions of a Phoenix" (1843), reprinted in The Works of Thomas Hood, volume 8 (1872):
In the meantime, the ten o'clock Clapham omnibus called for me as usual ; I put on my hat and gloves, took my walking-stick (the dead don't walk with sticks), got into the vehicle, seated myself, and remarked with a smile all round.
"Well, this is better than a hearse."
A speech natural and significant enough under my peculiar circumstances, but to the rest of the company, who wanted the key, a mere impertinent truism.
An early instance of the man on the Calpham omnibus being presented as a representative figure whose preoccupations are worthy of investigation occurs in a review of Henry Russell, The Letters of Civis on Indian Affairs, from 1842 to 1849, in The Calcutta Review, volume 13, number 26 (1850):
If you want to see a real man of business, you must go to England, not to India, for the sight. Look at a man, who takes the Clapham omnibus to Grace Church-street, every day at ½ past 9 in the morning, and takes it again, in Grace Church-street at 5. He returns home to dinner, tired, care-worn ; his mind in his ledger and his banker's book ; brooding over £ : s : d ;—very bad company for his wife ; perhaps cross ; perhaps sleepy. His "recreation" is an after-dinner nap. He has no holiday, except on Sunday. He does not know what amusement is. His life is an "eternal grind :" but he gets used to it in time, as the horse does turning the mill-stone.
Another early occurrence is in this item from the November 1, 1856, issue of Punch:
Charming Simplicity of an Elderly Lady from the Country
"They tell me the Coachmen and Cabmen are so much more civil in New York than in London. For myself, I must say that I have invariably found the omnibus conductors about the Metropolis the most obliging of men. For instance, I have occasionally hailed a Clapham omnibus by mistake, and inquire if it was going to Hammersmith, when, will you believe it, the omnibus-conductor has always said to me, with the most charming politeness, 'Jump in, Ma'am!' Now, supposing I had taken the poor fellow at his word, only consider how much he must have gone out of his way to oblige me!"
The Clapham omnibus also appears in Samuel Smiles, Lives of the Engineers, volume 5 (1879), where it stands as a model of punctuality:
The express meat-train from Aberdeen arrives in town as punctually as the Clapham omnibus, and the express milk train from Aylesbury is as regular in its delivery as the penny post.