The clicking noise that you hear is the starter solenoid trying to get the starter to run. Chances are the bendix in the starter is locked up. A good fix for a locked up bendix is a few good taps with a hammer. This should free up the bendix.
A clicking noise is not a common sound for a bad ring and pinion or axle bearings. Use electronic chassis ears (denlors ste6600) to help pinpoint the source of the noise. U-joints in the driveshaft could make a clicking noise although it should click or make more noise on acceleration. I know you said it didn't, but listen REALLY carefully, and if it's still there when you are hitting the gas it could be your struts or your wheel bearings.
check your fuse. i had the same problem. that might be it. I doubt a fuse, the clicking is probably the horn relay that you hear. Fair chance the horn or connection at the horn is the problem. The clicking noise is the relay, and since it is clicking I doubt that your switch is bad, so check the fuse and also check the horn (hook it to the battery).
I would have to hear the noise to be sure but it sounds like the attery could be weak. The clicking you here may be the starter solenoid trying to engage, but it cant, because of a lack of electrical power. If that is the case, the clicking will probably be a rapid clicking noise, that might sound like a woodpecker. The salanoid switch or the battery could be bad. Also make sure you have good clean tight connections. Sometime you have to disconnect the electric cables and clean them with sand paper. Then reconect them.
It is difficult to say from your description, but it might be either a bent rim, or a Tire with Ply Separation in the sidewall or tread. However, if there is only a noise, and no FEELING of the wobble, then I would not think these are the causes. In a NOISE ONLY situation, the sound could be from a bad wheel bearing.