Here is a page that describes some basic techniques for noise figure measurement: https://www.maximintegrated.com/en/app-notes/index.mvp/id/2875
The simplest technique, mentioned by The Photon, is called the 'gain method'. In this method, you need to know the gain of the amplifier (which you can measure with a spectrum analyzer and a signal generator) and the output noise power when the input is terminated. The 'magic number' for noise power is -174 dBm/Hz at room temperature. Take the difference between the measured noise power and the room temperature noise power (-174 dBm/Hz) and subtract the gain to get the noise figure. It should be possible to add some more gain after the amplifier in question to get well above the spectrum analyzer noise floor, then figure out what the noise figure of the first state is. Essentially, you build a relatively high gain amplifier, measure the gain and noise figure of that, then stick your amplifier on in front of that, measure the gain and noise figure of the combination, then use Friis' formula to get the noise figure of the first stage (F = F1 + (F2-1)/G1 + ...).