So I made a coffee mug in Blender, and want to add it to Morrowind, but I have no idea how to save it as a .nif!
Could someone help please?

I have these related programs:
Paint.net
Nifskope
Blender
TES CS

Download all the things http://niftools.sourceforge.net/wiki/Blender, the NifScripts, PyFFI, and Python are all you should need. Nicscope included for making sure the model exported properly. Also make sure you're using Blender 2.41 as future versions aren't compatible with the scripts linked above.

AHA, thank you StarWarsGuy, I'm using 2.66a.

Just also, will I need to do anything with my model other than export as .NIF then incorporate it into Morrowind? Remove camera/light source? Change size somehow?

Import a vanilla cup found on the CD, I have the goty edition, but I know some people have a separate CS CD... Or even steam, which may be a problem as you'll probably want to extract the BSA somewhere... anyway: import than into blender then invert selection to reselect your cup then scale it to the same size.
If you mean the Blender light and camera, you should delete them, I usually do it before I make something.

Don't forget to export to the Morrowind format... it's caught me off guard a few times (and is why there's a faulty de_Table_03 in MSGO. My bad.)