If it focuses in bright light but doesn't in dimmer light, it sounds like the aperture diaphragm may not be opening back up all the way as it should after a photo is taken. This could be due to either a defective servo in the aperture unit or a problem with the connection between camera and aperture unit. If you've eliminated the contacts between the lens and body, then it may be a loose ribbon cable inside the lens. In either case, the repair should be left to someone with the proper training and tools.
One way to test this is to set the camera to M exposure mode and meter a scene with another lens set to f/4. Adjust the shutter speed and/or ISO until the meter is centered. Then meter the same exact scene with your 24-105 set to the same focal length/field of view. Is the meter reading differently between the two lenses? If the 24-105 is metering one or more stops darker (i.e. requires a slower shutter speed or higher ISO to center the meter), then it means the aperture is not opening all the way up.
If you have IS turned off do you get the same noise when you half press the shutter button? What you are hearing may be the IS unit starting up.
So it seems it is time to send the lens to a service center. With the recent decrease in the cost of a new EF 24-105mm f/4 L IS (around $700 in the U.S.), anything other than the base charge for a lens repair will be an appreciable percentage of the cost of a replacement lens.
In the case that you decide the lens is a total loss and you feel like trying to fix it yourself just be aware: taking a lens apart yourself voids the warranty. Additional warnings are included in this blog post by Roger Cicala of lensrentals.com. And all he is discussing in the case of the 24-105 is the much easier removal of the front element to clean out dust. Getting to the aperture assembly in the 24-105 involves removing at least one set of 3 eccentric setscrews that are used for the alignment of the 3rd group optics in the aperture assembly. If you don't have a way of checking the calibration that these three screws adjust when you reassemble the lens, you will get very poor optical performance.