Todger, from northern English tadger
The OED says todger is British slang, a variant of tadger, and their first citation is 1986:
1986 Comic Relief Christmas Bk. 135/2
Shakespeare uses Comic Relief..to relieve the
audience from tragedy with cunning allusions to
the enormous todgers that were the joy of his
For tadger, they say it's British slang originally from northern England. The etymological origin is unknown but they point out their first quote and a later quote:
1949 E. Partridge Dict. Slang (ed. 3) 1192/2
Tadger, penis: North Country, esp. Yorkshire...
Perh. ex tadpole.
1990 T. Thorne Bloomsbury Dict. Contemp. Slang
(1991) 506/1 Tadger,..a vulgarism of unknown origin (prob. from a lost dialect verb), used for
many years in the North of England and revived by
students, alternative comedians, etc. in the
1980s. Todger is an alternative modern version.
Tadge, to beat to a pulp
However, the New Partridge Dictionary of Slang and Unconventional English (2008) says:
todger noun the penis. From the obsolete verb ‘todge’ (to smash to
a pulp), the penis seen as a smashing tool UK, 2001
tadger noun the penis. Originally dialect; survives in rhyming slang
FOX AND BADGER UK, 1961
The obsolete verb todge is found in Slang and Colloquial English by John S. Farmer and W. E. Henley (1921, New York):
Todge. Stodge : as verb, to smash, to pulp.
And likewise in the Vocabulum; or The Rogue's Lexicon by George W. Matsell (1859, New York):
It's defined as a noun in Francis Grose's 1785 A Classical Dictionary of the Vulgar Tongue:
Todger, Yorkshire dialect for a child?
Todger may also have been Yorkshire dialect for a child, judging by the title of a poem by Gwen Wade (1904-1996), a "distinguished Yorkshire Dialect Society member". Transactions of the Yorkshire Dialect Society - Volumes 15-17, 1984:
Modern Yorkshire dialect poets such as Bill Cowley in his "April Bairn" or Gwen Wade in her "T'Little Todger" come closer to black American dialect writers' treatment of children. They write almost exclusively about the delight children give us, or they write "teasing" poems for the children themselves (to scare them into obedience?).
Here's an extract:
By gow, but tha sewerly is cappin
Thro thi heead dahn to t' soles o' thi feet!
Ah could credit at t'Dobs caught us nappin
An slipped us a fresh un bi neet;
We'd waited on summat o' mettle
Bein born o' th o' thi mother an me,
But, peeakin up theer on t'langsettle,
Why, sitha, tha's thee!
A snippet of Mixer and Server (Volume 35, Hotel and Restaurant Employees' International Alliance and Bartenders' International League of America, 1926) uses "little tadger" to refer to a boy.
Seeing the little tadger trying to reach the bell and thinking he was a good scout and doing his daily good turn, he walked up to the porch, smiled at the youngster and said, "What can I do for you, young man?"
Perhaps less relevant, but a snippet of The semantic development of words for "walk, run" in the Germanic languages by Roscoe Myrl Ihrig (University of Chicago press, 1916) offers todgey meaning "short and fat":