I was called "Dick" or "Dickie" until I got to high school (1968). There was kidding about it, but not in a major way. Still I ended having people call me "Richard". Then as a sophomore a girl took me aside and said that "Richard" sounded way too formal. She suggested "Rich" or "Rick" instead.
Other names with the same connotation include Peter and Johnson. A "polite" word often used is member. Out of curiosity, I checked with the Ngram viewer:
interactive Ngram viewer (note that it is normally case-sensitive)
It turned out that "his johnson" was just crowding "his peter" at the bottom, so I left it out. My concept about Peter was wrong. I had thought it had been the popular substitute term, and that the graph would show the word peter fall as dick rose. (Sheesh! 😱 I know how that sounds, but I don't know how else to say it.)
After posting, it occurred to me that I should have included "his manhood", as the OP originally did. You can check for yourself via the link below the image. It turns out that in 2008, its usage fell between "his dick" and "his penis". But from 1820 on, it dominated "his penis" and did not fall below until about 1950. I'm not replacing the original plot, though, because "his manhood" has other common meanings, such as, "upon reaching his manhood". I suspect it was that context which allowed it to dominate the plot for so long.
Yeah, avoid using the name "Dick". "Rich" and "Rick" are both great.