Welcome to AAC.
The wavelength of the 2.4GHz signal is about 125mm. A ¼ wave antenna would be about 31.25mm. Lengthening the antenna is probably not going to help.
If you drill a hole in the bottom of the drone and feed the wire out so it hangs, then run four ~33mm wires from the PCB ground down each arm of the drone, that might improve things. The ground of the PCB will be the trace that the battery’s negative side is connected to. There is not much risk even if you make a mistake since there will be no connection to the other end of the wire (be sure it is insulated!)
Try to pick a spot on the PCB that is as close as possible to the antenna connection. One thing that might be worth trying is to do the first part of having the antenna come from the bottom and see if the negative leads on the LEDs in the arms work as is.
What you would be doing it creating a ground plane with the four radials. It‘s also called a counterpoise. The purpose is to provide a ground reference to the ¼λ* radiator, the name for the main wire of the antenna. The LED wires might work for this.
¼λ antennas depend on the presence of an image antenna which is a virtual copy of the antenna extending into ground. If the antenna is not actually close to the ground, a set of radials can be used to create 0V references for the radiator in the same way the actual earth does when the antenna is near to it.
The problem you are going to have is that antennas need to be matched to impedance of the transmitter or receiver they are attached to. To make this match, and antenna must be tuned. Usually this is done with instruments that can measure forward and reflected power. This is not practical in your case so you will have to “cut and try”.
33mm is going to be “close” to the right length for the radials. If you want to give it a try it could make a pretty big improvement but there is no guarantee. Any other modifications are likely to run afoul of FCC (or other agencies’) regulatory requirements. Even adding the ground place is dodgy but you won’t be modifying the antenna at all, so…
*λ is the Greek letter lambda and is used to stand for wavelength
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