[renlute]

What do the cable DIYers and pros use for the approximately 1-inch-long sleeve that protects the cable at the point it enters the ear cup housing, and helps avoid too much flexing at a given point? From illustrations it appears that there are numerous possible solutions.
 
Exterior diameter should be 1/4" to fit the housing. I'll be enclosing a cable of 1/8" diameter.
 
I've combed through quite a few threads and am uncertain what this part is called. Likely it's often trimmed from a much longer piece.
 
And then where to buy it in a small quantity (I should only need one). Digikey would be nice since I have another order being packed there right now.

 

[logwed]

That is heat shrink used as a strain relief (the exact same purpose that you guessed, not flexing too much and potentially breaking the connection). I'm not a big DIYer (read: at all) so I'm not sure what diameter you should get. Also, I'm pretty sure that heatshrink is sold by the foot, but it's pretty cheap.
 
(Do you notice that I use a lot of parentheses when I'm posting late at night?!?!?)