I think some of the things that haven’t changed - even in their method of manufacture - would win out. AFAIK, wicker baskets and some kinds of clay pots are still made the way they would have been in the neolithic, whereas knives, axes,wheels - even needles - have all changed in materials, form and/or manufacture.
However, hammers today are made of carbon steel, for the most part. That didn’t exist when they first started using hammers, so that constitutes a pretty significant redesign. Nobody uses stone hammers anymore, unless it’s for some antiquity-related purpose. Even remote tribes have gotten metal tools.
On the hand, take nets. Some nets are still made of natural fibres, like hemp or wool or what have you. Nets go back God only knows how long. Some are made of artificial fibres like nylon but many in use today are not. So that design hasn’t changed. (Rope, same thing.)
Mangetout also mnentions bricks. Many bricks used today are not substantially different from bricks of 5,000 years ago, made of mud or granite or what have you.
I recall in a Ripley’s Believe it or Not cartoon from a few decade ago that the Shadoof (which has various spellings) has remained virtually unchanged for many thousands of years. It’s a device that uses a counterweighted bucket to transfer water from one level to another, used in the Nile and close by.
Unlike other items listed here, this one was, at least until the past century or so, constructed and used in exactly the same fashion and from the same materials as when it originated. So no issues of “is the modern needle made the same as old ones”? (If that weren’t an issue, I think my vote goes for “needle”. Stone age needles are essentially the same design as modern ones – a long thin needle with an elliptical hole at one end, used for drawing thread.)
I agree with this. Acheulean (sp?) hand-axes were used 500000 years ago, but they’re worlds apart from a axe today, with a handle and a blade of forged metal.
For the most part, stone tools look too different from their bronze or steel successors that it’s hard to consider them the same.
But needles made out of bone look very much like modern metal ones, and the same is true of fishhooks. Thread must go back almost as far.
I’d have to say: Shillelagh.
Simple lightly-reshaped hunk o’ wood for whacking things, animals, people. And for walking. Certainly had to predate even the stone knife. Og the First would recognize it, and be able to use it.
Of course, we should consider the handle itself.
Stone tools take a quantum jump in sophistication when you add a handle. And, the handle on a modern tool is pretty much the same thing, and in some cases made of the same materials as its fifty thousand year old precursor.
Raw hide mallets are still used in metal working, and wood crafts today. Roll up a skin, tie on a handle, with strips of hide, and let it dry. Another fifty thousand year old tool design.
Tris
Ditches.
Right, but hammers are different today. There design has changed to increase their ease of use. Most people do not use hammers with wooden handles anymore.
The sickle has remained identical in shape and function for over six thousand years; the first ones were made of wood with embedded flint edges.
Grinding stones have been used since Neolithic times, and are still used in some cultures today for grinding grain into flour.
Obsidian worked into a cutting edge.
Such blades are in use today in surgery, but date back a long way, for use in surgery in Ancient Egypt.
Some Obsidian use is dated at least to the Upper palaeolithic around 11,000 years ago, however it certainly goes back very much further than that.
Time Machines?
Charo. coochie coochie! 
An anvil? Other than rust, how would you tell a new one from a 100 or 200 year old one?
OK, a mallet then.
Chopsticks or perhaps the boomerang.
Dugout canoes are probably high on the list. Still in use in Africa and in some Pacific cultures.
The lever?
What about walking sticks? A walking stick now is still just essentially a polished piece of curved wood.