I am trying to get a sample read out from a laser diode on an oscilloscope. I get a good pulse signal if I shine my keychain flashlight on the detector, but a very noisy signal from the diode itself. What could be causing the noise in this system?

It seems like you are experiencing some temporally-localized interference. If you look at a short enough slice of time, the photodiode might pick up the constantly shifting course of constructive and destructive interference that is basically what its called speckle.

Laser Diode Modules have a fair amount of noise if you look on a very short time scale. The total noise is a combination from multiple sources including but not limited to power supply noise, mode hopping, thermal effects, and optical feedback. These noise sources can be severely reduced by using a low-noise power supply and a heat sink or preferably a Thermo-electrically cooled (TEC) laser. In order for a laser to have a perfectly consistent output even on that small of a time scale, the necessary controls need to be designed in from the ground up, which is generally not the case with laser diode modules.