I know a guy who actually has a name for his penis and seemed surprised that I didn't.
Anyway, In polite company I'd use "penis", not that it'd ever come up in "polite" company... well, with a doctor... maybe.
"Cock" reminds me of preppie-like sorority girls with Valley-like accents going, "Oh my Gaawd! Did you see his cock???" ....
"He'd a French cocked-hat on his forehead, a bunch of lace at his chin,
A coat of the claret velvet, and breeches of brown doe-skin;
They fitted with never a wrinkle: his boots were up to the thigh!
And he rode with a jewelled twinkle,
His pistol butts a-twinkle,
His rapier hilt a-twinkle...."
taaa ta taaa ta taaa......
anyway...
In writing I jump to the quick and go with dick (HEY, that rhymes). Others have used the latter as well, as in "big dick _____", for the sole purpose of distinguishing me from other _____'s, no doubt, though first time I heard the term it took me awhile to figure out to whom they were referring.
My personal preference is simply "dong". Dong. Whimsically irreverent yet as a descriptive, seemingly apropos.