Went out to start my 90 SSE and when I turn the key I get one click; does not turn over. If I do this 3-4 times it will fire right up. What I'm wondering is whether there is something in the starter that is freezing since it is below zero degrees here. I don't believe it is the battery. Anyone have this sort of thing happen and get it fixed to know exactly what it is? When I looked up starter parts I noticed there is a starter and a starter selenoid.

pull the leads off the battery and check for corrosion.

I'll try that tomorrow morning before I go to start it again. I don't think it is a corrosion problem because the battery isn't very old and there was no corrosion when I replace the old one but it is a good place to start. If that doesn't work I might replace the starter selenoid since they are relatively cheap. Do you know where it is located under the hood? My 90 SSE is along the 88-90 era.

The solenoid is a part of the starter. To change it, you will need to remove the starter. It is on the driver'* side, front of the engine. Remove it from underneath. Use search feature for specific instructions.

I don't think this is the solenoid. The starter would spin freely without turning the motor if it were.
Sounds like a low voltage problem or the starter brushes are worn past their limit. The repeated clicking jogs the armature to a good spot and then it turns over.
In that case you will need to R&R the starter or change the brushes as I did when this happened the same day as the NEBF.

The odd thing is that it only does it the first time I go to start it in the morning. The rest of the day it starts right up. If there was a worn part in the starter wouldn't it act up every time I went to start it throughout the day?

You could have an overnight battery drain happening or the starter going out.
You could check the battery voltage first thing in the morning to see if its low.
If it is try unplugging the power antenna in the trunk and see if it happens the next morning, its a common problem when they go out.

Well today is 20 degrees; a bit warmer. The first thing I did today after shoveling the drive was pull the positive and negative cables off and other that the rusty outer bolts they were clean as whistle where the actual connections are. I bolted them back up and started the car on the first attempt of the day. Although it did crank over 3-4 seconds before starting. My power antenna doesn't work so unplugging it probably won't help the amp power. I had a faulty ignition module last fall and I wonder if that might have impacted the integrity of the battery at least for the below zero degree starts. I'll keep an eye on it but I really think that a starter going bad would effect the starting of the car everytime I go to start it and like you said that it setting outside overnight is compromising the affect of the battery. I appreciate your help and will keep you posted if the problem becomes worse.

Your power antenna is "Hot at all Times" until it reaches the bottom limit switch to cancel the circuit.
If it is stuck and still connected you will drain amps when you turn the radio off.
Try this,
Place a battery trickle charger on it for about an hr before starting it in the morning.
If it starts right up, you have an overnight battery drain happening.

Putting a battery charger on will warm the battery and significantly increase its current output. This will occur as a function of battery temperature and allow for an easier start even if there is no battery drain problem. I have used this method on cold nights for lots of vehicles when I have been too lazy or cheap to install a decent engine heater.