I have a sticky throttle cable on my '87 VFR700. I took things apart and found what appeared to be dry graphite under the twist grip, and maybe also in the cables themselves. At any rate the cables were dry. The stickiness appears to be in one of the two cables, not in the twist grip. I looked on the internet and found some people saying to not use any kind of wet lube on cables if they have teflon interiors. For example: Link: To oil clutch/throttle cables or not? The owner's manual says nothing and I found nothing on this forum. Any opinions? How does one tell if it is a teflon cable or not? For what it's worth, I'd probably prefer a dry lube since I live in a fairly dusty, windy place. Rust is not a problem.
If you decide to lube them, you might try this; DuPont Multi-Use Lubricant with Teflon Fluoropolymer at BikeBandit.com I have also used "LPS #1", and the stuff in the green can I can never remember the name of. It's either "Cable Lube" or "Cable Life". Both of these are aerosols, so they are messy. The product above is a drip-on application. Like RedRover points out, your cables are old. It would be worth looking closely at the ends to make sure they aren't ready to give it up.
Actually, the cables look quite good. Maybe been replaced already. Well, I just threw some motor oil in there temporarily. Guess I'll find out if they were supposed to be run dry! I will look at those products later. BTW I found the source of some of the friction: the bar end weight. I guess the throttle just needed to be positioned a bit higher on the bar!
I recently had the same problem - the rubber grip was getting hung up on the bar end... a couple of pulls toward the center and it was fixed. I am pretty certain, though that your experiment with motor oil isn't going to end well... the oil will attract dust and gunk up the cables pretty quickly. I'd recommend removal of the cables, followed by a a dousing with brake cleaner to get the oil out. Then use one of the dry lubes suggested by Lgn001.
