Most of the webcomics that I follow have a good storyline but I don't mind some randomness. When it comes to a more random story a good sense of humour helps though. Most random comics that I have seen have been humour so I'm guessing that is what you are aiming for? The art doesn't have to be amazing but something distinctive about your style helps to make it memorable.
If you are interested, I would say that my top online comics are (and some reasons why):
Ctrl+Alt+Del (Humour and sometimes plot)
Earthsong (Plot, art)
Head Trip (Humour, when it was updating anyway...)
Looking for Group (Richard! XD)
Muse Academy (Plot, big emphasis on the Greek theme in the art)
Questionable Content (Humour, and sometimes story too.)
The Dreamer (Art, plot)
The Phoenix Requiem (Art, plot)
The Zombie Hunters (Zombies! Has experimented with the art form over time.)
Weregeek (Geeky humour and sometimes there is a story arc)