hvy barrel has provided a good overview.
First thing you need to do is get the nomenclature correct. That piece on a 350 is properly called a harmonic damper. As such, it has absolutely nothing to do with balancing the motor. It is there to cancel out vibrations, or harmonics, that are generated by the twisting of the crankshaft due to power pulses from the piston. When the plug fires and jams the piston down in the bore, the rod throw on the crank twists out of shape a little. When it has sprung as far as it is going to one way, it springs back the other way as far as it can, then reverses and springs back the other way. Think of it as a plucked guitar string. If these harmonics are left uncancelled, they will eventually crack the crank and it will break (at least 50 miles from home when you have $2 in your wallet).
It is constructed with an inner hub and an outer inertia ring, separated by an elastomeric band of material which allows the inertia ring to rotate slightly clockwise and counter-clockwise in relation to the inner hub. On all internally-balanced motors like your 350, there is no eccentric weight on the damper. On some other motors, like the small block 400 and big block 454's, the motors are externally balanced with eccentric weights being placed on the harmonic dampers (now called harmonic damper/balancers and on the flexplate or flywheel.
There is a veritable plethora of different dampers for the small block Chevy. In addition to different thicknesses and diameters, there are also 3 different TDC ignition timing locations cut into the outer inertia ring. Unless you bought the car new and have owned it all its life, you have no idea if the damper and front cover (where the timing tab is located) are the same ones that came on the motor from the factory.
Fellows who are looking for immediate rpm gain, such as coming off the corner in oval dirt track racing, might choose the thinnest, smallest diameter damper they can find. Other fellows, who are building a streetable motor that they want to last for a while, might choose the largest diameter and heaviest damper they can find because it would be thought that such a damper would afford additional protection to the crankshaft.
I have used these guys for decades. They can provide you with a rebuilt damper in exchange for yours or you can buy one of their rebuilt units outright. They disassemble OEM dampers and damper/balancers and press them back together using new elastomeric material with hydraulic pressure. Once a stock OEM damper has been in service for a while, the elastomeric material begins to degrade and will sometimes allow the inertia ring to move in relation to the hub. This renders the TDC notch ineffective and any further attempts to time the motor with a timing light will be fruitless. I like to use these guys because I'm still dealing with an OEM hub, which I know is the correct diameter to provide a tight press fit onto the crank snout. Some aftermarket units are oversize, do not provide a press fit and will not be effective as a damper of harmonics. The unit must be tight on the crank in order to transfer the vibes to the inertia ring.
Click on harmonic balancers on the left of the page, then click on Chevrolet & GMC.
Like I said, there are 3 different positions for timing tabs and dampers on a small block Chevy, 12:00 noon, 2:00 O'Clock and 2:30 O'Clock, You must find TDC on #1 cylinder and install the proper damper with the proper timing tab in order to be able to time the motor with a light.
Also, let me help you guys out with this warning:
DO NOT HONE THE HUB OF THE DAMPER SO THAT IT WILL SLIDE EASILY ONTO THE CRANK SNOUT. IT MUST BE A TIGHT PRESS FIT OR THE UNIT CANNOT TRANSFER HARMONICS TO THE INERTIA RING TO BE CANCELLED OUT AND THE UNIT IS THEREFORE JUNK.