Sorry hadn't meant to be so abrupt, but was on my phone and something came up. The point is that its hard to offer advice about something like this if we don't know what your goal is for it.
Sorry, as the op I should have answered you earlier
One reason is simply a matter of conversion from older versions of D&D. In most of our previous games we have had a character that has done some form of shield bashing or spiked shield use. In particular 3e had very specific (although awkward) rules about attacking with a shield though I have an old AD&D character still that attacks with a shield
However the main reason is that I'm running a campaign set in and around Amn in the Forgotten Realms and I wanted to introduce some alternative weapons and armour to emphasis the conquistador style of that land. I have some other ideas too, but was a bit stuck on the shield
My general rules of thumb are that I only really want to introduce new equipment or modify existing stuff in a limited fashion
I'd like this equipment to be of use without class abilities and feats, though possibly better with them
I don't want to modify any feats or class abilities except for possibly adding limited new uses for them e.g dual wielder and a shield bash.
I want to keep the equipment as simple as other 5e equipment. So for example although a bucker might be best represented by having a differing AC bonus vs melee and ranged, or even being used as a reaction rather than a generic AC bonus, I'd rather not go there.
So far based upon several posts suggesting that a shield should be about as damaging as an unarmed strike I am in agreement I think. Well, I am when referring to the 5e shield which is basically a 'large' shield like a heater or kite shield. The size and surface area would make it unwieldy and do limited damage. As Ancalagon said above it would be better for shoving than damaging.
So a phb shield could be used as a simple melee weapon, doing 1 point of bludgeoning damage + Str. This could be used instead of a regular attack if unarmed or needing to do bludgeoning damage. With the dual wielder feat it could be used as a bonus attack like normal. I'm heading towards allowing the +1 AC too since you get no other real benefits of that feat, except for being able to equip a shield and draw a weapon as a single action. It basically makes dual wielders good with two weapons, or weapon and shield
Then I add another shield in which I'll call a buckler (completely unrelated to the totally fabricated 3e buckler). Now these were the ones that were historically used to attack with and so should do more damage as the buckler is designed as a weapon as much as a shield
+1 AC being a shield plus light weapon 1d4 + Str bludgeoning damage. Can be used as an offhand weapon without the feat since its light, along with another light weapon. Now +1 AC is probably stronger than an extra point of damage from a 1d6 weapon or to the option of range from a dagger or light hammer but I'm ok with that since two weapon fighting is a touch weak anyway
Next up I'm thinking of creating a few interesting codified options to do with the free hand from the dueling class ability since with my shield rules it doesn't work. Same applies for versatile weapons.