Readymade "DuPont" style wires with female connectors on the ends, that mate to male 0.1" headers soldered on your board. That's the mess on the left part of the picture. These are also great for probing stuff, as it is possible to install a male header pretty much anywhere you can solder, and that solves the three handed multimeter problem.
For more robustness, the mains version: spring-loaded wagos. It's very easy to modify the wiring, and the high voltage bits are hidden inside, which is safe.
Regarding your exact question, from an "ease of prototyping" point of view:
If you want to remove the wire, then put a new wire back in, then the problem will be that the hole is full of solder. You can always suck it out with the solder sucker, but if the hole was tight to begin with, the new wire still won't want to get in.
The solution is to make the hole too large for your wire. For example, 1.5mm holes for 0.5mm wire. This makes it much easier to experiment: just melt the solder and push the new wire through the melted solder blob. It will find its way through the hole very easily. Then angle the wire 45° so more of it sits in the hole, and finish soldering it.
That works very well. These SMD pads stay attached to the board in spite of the heavy transistors with heat sink because I put in holes inside the pads. So they're SMD pads, but with a via in the middle. You can also put vias in the corners too. The vias take up less space on the other side of the board than a full thru-hole pad. The solder fills the via and holds the pad to the board.
Then you can make a hole, and thread the wires through it. That will hold them and prevent them from wiggling until the soldered end breaks. You can also make 2 holes, slip a ziptie through, and hold a cable in place.
It's not pretty, but it's nice to be able to mess with the prototype without the wires periodically coming off.