Mine are not currently up to what I want them to be, but that is mainly a funding thing. The following is pretty much my opinion, and I might be biased, but I think I'm right.
First, there's a difference between something that is "better than nothing", something that is decent for the job, and something that is specifically designed with HD in mind. What I currently own I think falls in the middle.
Size - you want something as small as can be, while still being legal. Most "tactical" shotguns have an 18.5" barrel length. This makes it easier to navigate through your home if you have to, and gives you more places where you can store it.
Capacity - you might as well get the tube flush with the barrel. This generally means 6-7x 2.75" shells in the tube (depending on how far they're crimped, I have some "2.75 inch" shells that look more like 3" ones), or you could get a box-fed. Extended tube came standard on my mossberg 930 SPX, I had to buy one for my Benelli Supernova.
Along these lines, a pump action is generally more affordable, but I think the advantage lies with semi-autos. Mossberg 930 SPX with ghost rings and a top rail, straight stock, and extended magazine tube was $640.
Stock - personally, I prefer the straight stock on a shotgun, especially with a tube magazine. Makes them fit places easier and I think it absorbs recoil better. You want a shorter LOP than you would with a sport gun, to shorten the overall length.
Accessories - side saddles are nice, but not really necessary if you have a decent capacity. I have one on my benelli because it is part of the top rail. A flashlight for target identification is a very good thing to have, it prevents you from shooting your kid when he sneaks in late at night, thinking he's a burglar. There are different mounting options for different shotguns, most use a barrel clamp. I recently purchased a sling, which makes it a lot easier to carry around if you have to.
The last thing, which I do not have yet for either of my shotguns, is a red dot sight. Yes, you can do just fine with others, but I think the faster target aquisition of a red dot can give you an advantage (the difference between "good for HD" and "designed for HD"). However, you want to stay away from cheap red dots, as I've had one (that claimed "rated for .50 cal") break after less than 10 shots.