What qualities to look for in a cut of beef destined for "well done"?
I'm rather surprised by the judgmental tone in some of the answers here.  A well-done steak is a culinary preference; just because you don't share that preference is no reason to be rude about it.  Some people like caviar; others don't, despite the fact that it is expensive and lauded by many "people in the know."  Some people appreciate an espresso made lovingly with freshly ground coffee in the "right" kind of grinder; others find it too strong or bitter and would prefer an "American style" of coffee with cream and sugar. Taste is subjective.

I'll admit something personal -- for the first quarter-century or so of my life, I only ate well-done or medium-well steak.  It's how my father always cooked it at home on the grill.  He didn't actually prefer steak well-done: when we'd go out to the restaurants, he'd generally order medium-rare.  But even though he'd time steaks, adjust heat on the grill, etc., the vast majority of the time, they were well-done.  I was used to it.  I liked it, because it was what I knew.  The few times I encountered less-done meat, I found the texture odd or even slightly off-putting.

Then, at some point, I was convinced to try more rare steak, and I soon accepted it.  I now almost always order steaks medium-rare, and I do prefer them that way.  But I also spent a lot of time ordering steaks more well-done (at restaurants, I'd almost always order medium-well), and to those of you who claim you can't tell the difference when a steak is that done, you don't know what you're talking about, because it's not your common way of eating and perhaps you've never had a well-done steak that was prepared in a reasonable fashion.

Anyhow, to answer the question: as some others have hinted at, choose a cut that has a "looser" texture with fat running through it if possible.  Also, consider using a cut that you'd often tend to slice thinly when serving anyway, like skirt, flank, bavette, etc.  You can also use more expensive somewhat fatty cuts (like ribeye), though the meat will toughen, so you won't get the benefit of the tenderness in such expensive cuts.  Plus some of the cheaper cuts (or at least less expensive) have superior flavor.  

Quality is actually more important in cooking well-done steaks, because older or worse quality meat with more connective tissue will become even chewier and tough when cooked longer. A somewhat high-quality ribeye cooked well-done can be a somewhat chewy but very pleasant caramelized experience with melt-in-your-mouth browned crispy fat interspersed.  A poor-quality steak with poor-quality fat will just become tough and have its bad qualities exaggerated.

What you want to avoid -- unless your guests insist on them -- are lean cuts, which will end up tough AND dry.  Filet mignon is a very poor choice (which will end up tasteless and tough), as is sirloin, as would be other lean tough cuts (like round).  Also avoid cuts with a lot of connective tissue (but sometimes are sold as steaks to be cooked fast and rare), like chuck.

Marinating will help if you allow enough time for brine to soak in a bit (thereby adding not only flavor, but more moisture).

As for cooking, keep in mind many types of meat are cooked to "well-done" temperatures and still can remain juicy with proper technique (e.g., chicken).  There's absolutely no reason to serve a tough, dry well-done steak unless you're incompetent.  A "loose" textured steak as mentioned above that's sliced thin before serving will be chewy but won't necessarily seem "tough" if marinated and cooked properly.

How to cook properly?  Do NOT do what most people do when cooking steak and just flip once.  You'll end up drying out both sides of the steak by the time cooking is finished.  Butterflying (which restaurants will sometimes offer to do for you when you request a medium-well or well-done steak) can be counterproductive for some cuts and can also dry things out more.  (Restaurants do it mostly for their own convenience; it speeds cooking.)  Unless you have a very loose-textured steak, which might benefit from additional browning reactions with greater surface area while not getting tough, you probably don't want to decrease thickness deliberately.  Instead, keep moisture in with a somewhat thicker cut.

And flip often during cooking.  It's more work because cooking to well-done takes longer, but it's the best way to keep juices moving around inside rather than boiling out the top while the bottom gets dry and burnt.  (Think of what a rotisserie does; you're doing the same by flipping steak often.)  Flipping frequently also can help soften fat and begin to break it down, which can add flavor and a "moist" aspect (if done right with high-quality meat, the fat might even be almost "melt-in-your-mouth").  Frequent flipping also aids in more browning reactions, which develop more flavor, and well-done steak does at least get that advantage of extra browning flavor (perhaps even crisped brown outer layers of fat).  Obviously control heat; overall you'll need to cook at a slightly lower heat to avoid burning the outside before interior is well-done.

Then pull steak off while slightly less than well-done, and let it rest to creep up to well-done.

To summarize:


Loose or well-marbled cut, cheapest are those you'd generally slice even if serving rare
Marinate for at least a few hours; salt will help with moisture, acid can help at least keep the outer layer less tough
Relatively high-heat sear on both sides
Then move to lower heat, and flip frequently
Check internal temperature with an instant-read thermometer to pull it off precisely as the pink is about to go away in middle (this may take some practice to find the right temp based on your cooking technique, how thick the steaks are, etc.)
Let rest to "coast" to well-done


I've accidentally overcooked skirt steak this way a couple times, and it was just as tasty (if not more so), basically as juicy, and almost as tender as if I cooked it to medium rare.  (Skirt is always chewy anyway; well-done steaks will always be chewy, but they don't have to be excessively tough.)

Cooking steak to medium-rare and getting passable results is relatively easy with a thermometer.  Cooking a decent well-done steak takes a lot more skill.

EDIT: A couple comments have noted that there are better ways to cook steaks.  I absolutely agree.  The OP didn't ask about preparation technique, so I was assuming a somewhat "standard" cooking technique for steaks (grilling, pan-frying, broiling, etc.), which my advice applies to.  Personally, I'd recommend things like reverse searing or finishing in the oven, etc. too for better results, but I wasn't trying to turn this answer into "how to cook a steak" in general.