Write the music or the lyrics first, but it really doesn't matter. If you write the lyrics first, then try to use symbolism and various phrases and words that don't seem to go together. The whole purpose of a random song is to keep the audience wondering what you are trying to say. The more obtuse, the more people will think you are being deep.
Add in chords that don't seem to go together either. Some of the coolest jazz includes random notes and riffs that don't follow in the same key or in a standard pattern. You want to stand out and wake up your listener's ear with a sudden break in standard musical protocol.
Create an aura about your music with words and music that is off the beaten path. As you are writing your music, if you hear another song in the riffs, take out one of the chords. Skipping around randomly is another way to create a cool but random song. Standard rock and roll riffs can be so boring, so play them out of order instead.
Put words together that do not usually go together such as rain and lips or trees and crutches.
Play with your song writing and try to shake yourself of any formal training that you've had that could stifle your creative random songwriting.