I've seen hand drills hanging on walls in restaurants and at rummage sales.
Is there still a place for "hand drills" in a modern woodshop?
Not so much in woodworking, but there are times when my wheel brace is my drill of choice -- when I'm much more concerned about control than speed, especially on thin, slightly flexible plastic. Here's an example: drilling drainage holes in a compost bin. A cordless, even on low speed, grabs and pulls through, a wheel brace doesn't.
I do sometimes use it for pilot holes in softwood, perhaps if the cordless has a clearance-size bit in it or is on charge. This tends to be for fairly rough work (e.g. joining 2x2s for a jig, where the 3-in-1 bits don't work because the holes are too deep). But I also tend to use spiral ratchet (Yankee) drivers rather than swapping a screwdriver bit into the drill (I used to have and frequently use some pilot bits that fitted into it via a 1/4" hex adaptor -- another form of hand drill).
A carpenter's brace (brace and bit) can still be handy for occasional woodworkers using flat/spade bits, if the only available power drill is too fast
This is the type of question that unfortunately tends to yield primarily opinion-based answers, but I think it's one of a number of exceptions to this being a bad fit for SE.
My answer is a definitive yes. Not only do I think hand drills have a place in the modern woodworking shop — regardless of the dominance of power tools otherwise — there are many cases where they should be preferred. You could argue I am slightly biased because I have a few and use them regularly, but there are many sound reasons to use them in specific circumstances instead of a power drill (of any type, but particularly cordless drills).
Just as with hand tools more generally two of the main reasons they can be a good idea are they can be used where no electricity is available and they are much quieter than the power-tool equivalent (virtually silent in many cases as the only sound heard is the bit working away in the wood).
Additionally. another common hand-tool argument: they are usually much cheaper. Vintage and antique eggbeaters and swing braces of excellent quality are available on the secondhand market to many in the West for less than their modern counterparts, and are incidentally better than nearly anything currently manufactured1. And needless to say they are much cheaper than most power drills2, just a fraction of the cost of certain name-brand drills (a tenth or less)!
Obviously not everything is in favour of the hand drill and the high speed of powered drills will generally give far faster results, particularly in hardwoods. And with power drilling some bit types will yield cleaner entry holes in wood (and note that certain modern bits are definitely most effective in power drills, Forstner bits being one example). However, drilling by hand is one of the few cases where the hand-powered version of something can actually be more powerful.
With a swing brace you have far more torque available to you, with maximum torque considered as virtually unlimited by some sources, depending on brace type and the strength of the user. So plainly more than almost any portable power drill can provide, but especially battery models for obvious reasons. Particularly when using a purpose-made bit such as an Irwin or Jennings auger, sharp and with a lead screw in good condition, a sizeable hole (1" / 25mm and up) can be made in strong hardwoods such as oak with surprising speed, and without the back-breaking effort one might imagine. By comparison many powered drills will struggle to do the same hole once it gets to a certain depth, and some will simply stall and be unable to continue drilling.
1 Examples from boutique tool makers are the exception but they tend to be eye-wateringly expensive, and frankly it appears most are bought as an investment or simply to look pretty on the wall.
2 In the UK and parts of the US it's easy to come by both eggbeaters and braces in decent condition (fully functional, just with cosmetic issues) for less than a tenner, sometimes less than a fiver. And as numerous forum posts and restoration guides tell us it's not completely unheard of to snag £1 / $1 bargains because of lots of visible rust, or the seller not appreciating what they have.
In the 'modern' wood shop? not really. drill presses and cordless drills can do almost anything you would do with a hand drill, faster and easier with less training.
However, they do still have a place. Many people like doing woodworking with differing levels of 'old school' tools and some times it's just nice to have quiet tools in your shop.
I have plenty of old tools in my shop and I keep collecting more. Sometimes they just give you the look your are going for on a piece or give you a deeper feeling of satisfaction when you have completed it, giving real elbow grease to the product.
Hand drills most definitely have a place. They are
They provide more torque for larger bits.
You need several drills:
Harbor Freight has a set of six augers for US$17 with 1/4" hex shafts. Combine these with a sliding T handle breaker bar and a 1/4 socket, and you have a T handle for any auger. Don't forget a square birdcage awl to start screws and a gimlet or two. The exercise will do you good. This is wood working, not wood slacking.
One advantage that has been touched on is that the low cost of manual drills means you can have lots of them.
For fine work (as opposed to construction) I find myself usually using a clearance drill bit, a pilot drill bit, a countersink, and a driver. Using one electric drill and switching the bits out got old really fast.
Now I have one eggbeater with a countersink permanently installed, another eggbeater for pilot holes, a short-throw brace for driving screws and my electric drill for the clearance holes. Doing that with electric drills would be expensive and would mean a lot of battery management. Total cost for all my manual drills was about £5 at a car boot sale.
Another issue with electric drills is that for fine work you generally want your holes to be very straight. Some people, including me, find it very hard to make a perpendicular hole with an electric drill. A drill press is ideal of course but is not always practical. On the other hand, eggbeater drills and braces form a natural straight line from the tip to the base of the handle that makes it easy to judge perpendicularity.