An old chestnut, but one not discussed for a while so here goes.
Firstly, it's not ecomonically practical. At least three Australian states and several US states have looked into this, and found that the cost of administering the system is higher than any revenue it could reasonably be expected to generate. Given the relatively low popularity of cycling, charging a registration fee on cyclists simply wouldn't generate the funds to cover the costs involved. Contrary to popular belief, this is the first thing politicians look at when deciding whether or not to implement a new tax (which is effectively all this would be). If cycling were to magically become more popular, it might be a possibility, but until then, it isn't going to happen.
Secondly, cyclists already to contribute to the upkeep of the roads through the taxes they pay that already go into the pool of general revenue -- the same place that fuel taxes and registration fees go. What the funds are then used for is decided by the government of the day, usually on the basis of where they feel the need to buy a few votes. Some of it goes to roads, and some to other things. The point is, considering the relatively miniscule portion of the roadway that cyclists use, it's fair to say that the cyclist has already paid for their share of the road regardless.
Finally, what exactly would you hope to achieve by suggesting that the cyclist should pay a fee? Paying an additional fee won't actually improve anything at all for cyclists (especially a fee that would be drowned in adminstrative costs). Motorists will still treat cyclists like crap, the police will still refuse to investigate cyclists' complaints, bikepaths will still be useless, and your coworkers will still find plenty of "reasons" to want cyclists off the road. You're far better off just to let them get their little bit of whining off their chest, and get on with your life.
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I am clinically insane. I am proud of it.
That is all.