I am guessing if you are riding a '69 Bonneville, you really are interested in what people have to say about the bike and aren't expecting a one line answer that cures all your vibration concerns.
The '69 is still English (right foot) shift, and has the original frame that the Triumph purists love. You probably know that - unless you have never ridden the bike. Most of us who have a Triumph work on them, and I am guessing you do as well, unless you recently inherited or bought the bike on a whim because of its classic look and name.
There are various vibration issues you have to deal with on a Triumph Bonneville. Road vibration, running gear vibration, and engine vibration.
Quick rundown - Run tires at proper pressure. Make sure your wheels are both round and true, and have them re-spoked or replaced if necessary. You would be surprised how a pot-hole can distort a wheel. Check chain and sprockets for wear, replace if necessary. Use proper lubrication. There are modern o-ring chains that are cleaner than original. Make sure your chain is not chafing against the guard. A surprising source of vibration on the Bonnie is the primary chain between engine and transmission. Make sure it is in good condition and tightened to proper tension. If you have to replace the primary chain consider a kevlar toothed belt replacement. They are quieter and smoother. Make sure brakes are true. On the '69 you *might* want to consider upgrading the front brake to a disc system. It will serve you better than the original double leading shoe. Make sure front forks are properly filled to proper level. You can go with aftermarket progressive fork springs, if you want to change suspension for more aggressive riding or higher loads. Also, you can replace your Girling rear shocks with aftermarket. I have had much better ride and handling with aftermarket products. In general, the Brit riders of the 50s and 60s were somewhat smaller guys than we are now. You *might* be at a fighting weight of 138lb, but I am guessing not. On later years, handle bars were rubber mounted, and ends of the bars had lead weights to dampen vibration. Again, you might want to consider that upgrade.
All of that is standard advice for road and running gear on any bike, with a few Triumph specific bits. If your bike is a "runner" you will think of it as modernized and tuned up. If your bike is a "shower" some of those mods might be too modern, and spoil the show value of the bike. Most are invisible - with the exception of the break.
More specifically about that particular bike - vibration really is part of the nature of the original British 360 degree vertical twin engine.
The 360 degree twin, in the T120 and T140 will ALLWAYS vibrate. Not up and down, but front to back. I know that sounds weird, but the crankshaft has counterweights that weigh the same as the pistons and connecting rods. As the pistons go down, the weights come up, and vice versa. So, the engine is balanced in the up and down plane. But, nothing is going back and forth to counterbalance these weights in forward and backward motion. That is the fundamental source of engine vibration. It won't go away. Ever. In the modern reincarnation of the Bonneville they incorporate a shaft, with a weight on it that spins to counteract that motion - vibration was enough of a complaint that the redesign addressed it specifically.
I have a T140. It does shake, but over 2500 rpm or so it isn't so bothersome. The engine in the Bonnie is rigidly bolted to the frame, not rubber mounted.
Things you can do to help with engine vibration? There are tuning issues that can increase vibration or make annoying vibrations at different rpm. If your engine is in tune and running well, and you STILL think it vibrates too much, trade the bike for a 1960s BMW R60 boxer.
Things that can cause non-characteristic vibrations? Make sure both carburetors are tuned exactly the same and throttles come off idle at exactly the same time. This is an annual tuneup item. Also, replace the points based ignition with a Boyer electronic ignition kit. More precise timing spark and more consistent spark will make for smoother running. That is a one time upgrade and it shouldn't break the bank.
If you are really, really concerned about vibration you can make sure you have stock camshaft, run lower compression pistons, and replace stock carbs with more modern CV carbs - Mikunis are common. If you do that, you would lose a lot of the character that makes the Bonneville a great bike. They aren't a Honda 90, or a BMW boxer. They are raw, require kick start, are messy and high maintenance, and have huge variation in build quality.
In the end, there is something about the Triumphs that is magic. Those guys knew how to turn $10 of petrol into 120 miles pure joy. No other bike feels quite like a Bonnie. Some vibration is part of that feel.
There are folk legends of 650s and 750s being disassembled, shipped off to mythical machine shops for "blueprinting" and balancing, but I have never see such a thing happen in 40 years of playing with Triumphs. Fundamentally, the design IS going to shake, unless you redesign it and put in a counterbalancing shaft.
Enjoy the unique character. IF you really want to stop the vibration, just turn the ignition off!
Triumph Progressive Fork Springs for T120 T140
1966 Triumph Bonneville T120 Carburetors Universal
331 Series Boyer Electronic Ignitions
Shock Absorbers For Classic British Motorcycles
Hayward Belt Drive Kits
530 ring Chain
