What kinds of (effective) melee weapon could only be made with modern technology? While we could make melee weapons with more modern technology, all we have are advancements in metallurgy applied to medieval era weapons. As no military wants melee weapons due to there being no reason to with guns existing, there hasn't really been any demand for a melee weapon designed using modern technology. While we could make melee weapons with a more advanced design no one is interested in trying to get melee weapons made with more advanced technology. So what kinds of melee weapon could be made in modern day if there was significant amounts of money & design effort were put into making an advanced melee weapon?
There are a few real life, common melee weapon that can only be made with relatively modern technology:
While not technically a "modern" invention, these weapons did not actually come into being until the 1800s: well after melee weapons had lost most of their utility on the battlefield. Although spring steel existed in the late medieval period, these knifes rely on being stored with their springs under tension and are held together by relatively small parts which make them very fragile compared to other knives. It was not until the industrial era that spring steel got good enough to make this sort of mechanism practical.
While the idea of a stick with a small weight at the end of it is one of the oldest weapon concepts in history, the ability to make this into a weapon that collapses into itself it fairly new. Telescoping Batons were only invented in the 1970s. The reason is because telescoping mechanisms have to be very precisely machined to be able to open up smoothly, and even more precisely machined to be able to lock itself into an open position when in use so that it does not just collapse on you. Furthermore the steel has to be of an exceptional quality or else it will warp when you hit someone with it which could cause it to come apart or bend in a way that you can't close it again.
Also invented in the 1970s, I think it's pretty self explanatory why these are modern only melee weapons. Even a light tap from a taser is enough to put someone on the ground making them one of the easiest melee weapons to disable an opponent with ever invented.
Strangely enough, flashlights are often considered by many experts to be the best self-defense weapon in the world. Not only are they non-threatening enough to carry around into most places, and often have enough heft to make it a good light club, but shining some of these newer generation flashlights into someone's eyes at close range can be quite debilitating without the risk of harming yourself like you get with tasers or pepper spray.
Conclusion: Role them all up into one package
Why settle for just one of these ideas when you can have it all. Make a collapsible spring loaded baton with a taser and flashlight at the end of it. It's portable, easy to deploy, has a bit of reach, has a lot of stopping power, and multiple ways to take out an opponent depending on the threat level... all without needing a lot of training or physical strength to use... oh yeah and the fact that they only cost \$75 and have non-leathal modes of operation are both really good reasons to own one instead of a gun.
Going past "modern"
A bit off-topic, but there are a few SciFi melee weapons, some of which could even be plausible, that could only be made with post-modern tech. The lightsaber is a cliche example; but a more likely, and possibly more dangerous weapon would consist of a weight at the end of a monomolecular wire with a handle, able to cut through almost anything by concentrating macroscopic force on individual molecular bonds. This melee weapon features in Larry Niven's Ringworld series---Larry's work is full of intriguing material science inventions.
Another melee weapon that could work in the near future is the vibroblade: a metal-bladed weapon, typically a knife, that vibrates at a very high frequency, "sawing" through the target material. Modern turbine-driven dentist drills work on this principle, except that they rotate very fast, instead of vibrating.
The principle of electromachining also comes to mind: in RL, an electrode tool is used to "eat" into conductive material by eroding it electrically, essentially by extra-fast ionic corrosion. If these tools could be speeded up by a factor of a hundred or so, they could potentially cut through armor (or robot plating) like a hot knife through butter. A waterjet could do the same to any material---such tools are widely used for precision steel cutting, but haven't been weaponized yet.
Edit: In Adrian Tchaikovsky's great Shadows of the Apt SF book series, in a world undergoing a war-accelerated Industrial Revolution, an inventor uses his sandblasting nozzle as a weapon of last resort during a siege. It is spectacularly successful, but the injuries it inflicts are so macabre, they drive the inventor to commit suicide in remorse. This weapon is not speculative; the tool exists in RL, and could readily be weaponized as is.