The Graze weapon mastery property states:

’If your attack roll with this weapon misses a creature, you can deal damage to that creature equal to the ability modifier you used to make the attack roll. This damage is the same type dealt by the weapon, and the damage can be increased only by increasing the ability modifier.’

BUT there are quite a few ways to add multiple ability score modifiers to an attack roll. For example Devotion Paladin’s Sacred Weapon:

‘When you take the Attack action, you can expend one use of your Channel Divinity to imbue one Melee weapon that you are holding with positive energy. For 10 minutes or until you use this feature again, you add your Charisma modifier to attack rolls you make with that weapon (minimum bonus of +1), and each time you hit with it, you cause it to deal its normal damage type or Radiant damage.

The weapon also emits Bright Light in a 20-foot radius and Dim Light 20 feet beyond that.

You can end this effect early (no action required). This effect also ends if you aren’t carrying the weapon.’

If a Devotion Paladin (or equivalent) uses Strength (18) and Charisma (18) in their attack roll with a Greatsword and somehow still misses, do they deal 8 damage?

Sacred Weapon wouldn't work in that case. Sacred Weapon only adds the bonus damage when you score a hit, graze occurs when you miss. Your own quote above states, "each time you hit with it" - graze isn't a hit, it is actually a miss that still inflicts damage.

It is for each melee attack with that weapon, and only the base state for a normal melee attack either strength or dexterity, nothing more.   The channel divinity and other bonuses you are talking about are only applies if you hit. Grazes are not considered hits. 

I’m not actually talking about hitting, but missing, you see the attack is both using the strength and charisma modifier for the attack (not damage) roll…

so Graze checks to see what was used in the to hit roll and then uses the modifier for damage on a miss…

in which case this is strength AND charisma

I mean, it seems actually fine. First of all, if you've built your character decently well and are facing sort of level appropriate challenges then you should be getting a decent number fewer misses anyway, meaning graze isn't doing anything for you. After all, if you haven't built your character well then this interaction won't be doing anything for you in the first place. 

I totally agree Wolf I just thought it was a really interesting interaction. If there is a decent build you can do with another version of this then it might do more damage to miss than to hit!

I think it is fine to run it as is treating both abilities as "the ability modifier you used to make the attack roll,"  but it is also easy enough to rule that the ability is strength (or dex) and the divine intervention just gives you a bonus equal to the other that would not transfer.

If you don't like shenanigans, don't have them in your game.

I love shenanigans, so yeah that would be how I would rule it. I’m just curious how far this could go and wanted some other people’s insight on the interaction.

Quote from Cheeky_Hamster >>

I’m not actually talking about hitting, but missing, you see the attack is both using the strength and charisma modifier for the attack (not damage) roll…

so Graze checks to see what was used in the to hit roll and then uses the modifier for damage on a miss…

in which case this is strength AND charisma

 No it does not grant both. The Charisma bonus of that skill  only applies on a HIT.   
Grazes are NOT hits, so do not get bonuses from abilities that are granted on hits. 
Graze only applies ONE base stat modifier for the weapon being used and that is it, nothing more. 
These are the RAW rules to Graze. 

IF you want to rule differently In Your Game, do so.  That would be a House Rule. 
But it is not RAW, nor RAI. 

Quote from Deyric >>

Graze only applies ONE base stat modifier for the weapon being used and that is it, nothing more. 

 That is definitely a way to rule, as I said. Charisma could be treated as a bonus to the roll, rather than "the modifier."

Quote from Deyric >>

 No it does not grant both. The Charisma bonus of that skill  only applies on a HIT. 

 I'm not sure what you mean here. The Devotion Paladin Sacred Weapon bonus is to attack rolls, not to damage. It couldn't possibly make sense to only apply on a hit.

EDIT: after reading the latest posts, I've changed my mind and I agree with @HarmAssassin

I may be misunderstanding something, but I'll post what I think are the relevant rules (snipped, with some added emphasis):

Attack Rolls: Ability Modifier
The Attack Roll Abilities table shows which ability modifier to use for different types of attack rolls.

Damage and Healing: Damage Rolls

[...]

When attacking with a weapon, you add your ability modifier—the same modifier used for the attack roll—to the damage roll. A spell tells you which dice to roll for damage and whether to add any modifiers. [...]

And finally:

Graze
If your attack roll with this weapon misses a creature, you can deal damage to that creature equal to the ability modifier you used to make the attack roll. This damage is the same type dealt by the weapon, and the damage can be increased only by increasing the ability modifier.

The issue here is "equal to the ability modifier you used to make the attack roll".
"The ability modifier". Definite and singular. You could use both if the description said "attack roll bonus" or "any ability modifiers", but it doesn't say that.
I'd say a nice DM would allow you to use whichever is higher, though.

You are of course right that the absence of a parenthetical plural opens the rule to be interpreted in different ways depending on individual emphasis, whereas the following would be clearer:

Graze
If your attack roll(s) with this weapon misses a creature, you can deal damage to that(those) creature(s) equal to the ability modifier(s) you used to make the attack roll(s). This damage is the same type dealt by the weapon, and the damage can be increased only by increasing the ability modifier(s).

However, this more open wording that allows for different edge cases and interactions is not how Wizards of the Coast write their rules - they write with the expectation of the most common interaction… Hence all the debates that then spring up around intention.

What remains in our books though is a rule that looks to use the ability modifier from the attack roll for damage on a miss… there is no assumption in the rule that it is possible to use more than one ability modifier for an attack roll and so the wording ought not be expected to accommodate such a wording - I believe then that if multiple ability score modifiers are present it is the clearer interpretation of the interaction to translate both to damage… I am of course biased though because I discovered this and thought it was fun!

Quote from Natrel >>

The issue here is "equal to the ability modifier you used to make the attack roll".
"The ability modifier". Definite and singular. You could use both if the description said "attack roll bonus" or "any ability modifiers", but it doesn't say that.
I'd say a nice DM would allow you to use whichever is higher, though.

 But if you follow that line of thoughts, then you were ruling you cannot stack Charisma with your Strength? And that is not correct.

Also, regarding plurals: 

Some features let you use different ability modifiers from those listed. For example, the Finesse property (see chapter 6) lets you use Strength or Dexterity with a weapon that has that property.

Damage Roll
A damage roll is a die roll, adjusted by any applicable modifiers, that deals damage to a target. See also chapter 1 (“Damage and Healing”).

Just to clarify my opinion: I'm not saying you can stack STR and CHA for the damage roll when you hit, only for the attack roll.

But at the same time, as I said previously, I believe you can add your STR and CHA for Graze.

EDIT: after reading the latest posts, I've changed my mind and I agree with @HarmAssassin

Quote from TarodNet >>

Just to clarify my opinion: I'm not saying you can stack STR and CHA for the damage roll when you hit, only for the attack roll.

But at the same time, as I said previously, I believe you can add your STR and CHA for Graze.

 I would argue that you can not, and it is because only one is an ability score modifier (STR in this case). The other is a bonus from your Channel Divinity that just happens to equal the amount of your CHA modifier. You were not using both STR and CHA modifiers to make the attack, you were using your STR modifier and Channel Divinity.

Quote from Deyric >>

No it does not grant both. The Charisma bonus of that skill  only applies on a HIT.   

Incorrect. The CHA modifier is added to the attack roll with Sacred Weapon, which is the entire point of this thread

When you take the Attack action, you can expend one use of your Channel Divinity to imbue one Melee weapon that you are holding with positive energy. For 10 minutes or until you use this feature again, you add your Charisma modifier to attack rolls you make with that weapon (minimum bonus of +1), and each time you hit with it, you cause it to deal its normal damage type or Radiant damage.

Add Modifiers. Add these modifiers to the number rolled on the d20:

• The Relevant Ability Modifier. This chapter and the rules glossary explain which ability modifiers to use for various D20 Tests.
• Your Proficiency Bonus If Relevant. Each creature has a Proficiency Bonus, a number added when making a D20 Test that uses something, such as a skill, in which the creature has proficiency. See “Proficiency” later in this chapter.
• Circumstantial Bonuses and Penalties. A class feature, a spell, or another rule might give a bonus or penalty to the die roll.

Strength is the relevant ability modifier and cha. is a circumstantial bonus.

Quote from Jurmondur >>

Add Modifiers. Add these modifiers to the number rolled on the d20:

• The Relevant Ability Modifier. This chapter and the rules glossary explain which ability modifiers to use for various D20 Tests.
• Your Proficiency Bonus If Relevant. Each creature has a Proficiency Bonus, a number added when making a D20 Test that uses something, such as a skill, in which the creature has proficiency. See “Proficiency” later in this chapter.
• Circumstantial Bonuses and Penalties. A class feature, a spell, or another rule might give a bonus or penalty to the die roll.

Strength is the relevant ability modifier and cha. is a circumstantial bonus.

Why? Are you being caught up with the third point because it is a (sub)class feature? So are many feature which would change (rather than add another) ability score modifier

1. Both Charisma and Strength are ability score modifiers

2. Both Charisma and Strength are explicitly added to the Attack Roll

3. Both Charisma and Strength are stacking for the attack roll and so they are relevant

4. If all the above are true then both are point 1 and are ability score modifiers to be calculated before proficiency bonus and circumstantial modifiers.

Even if the above wasn’t true, that is not the rules in question for this interaction.

You are completely missing the point. In your example, STR is the ability score used for the weapon. You are then using another ability to add an additional bonus. It is completely irrelevant where that bonus comes from (in this case it equals your CHA modifier). STR is still the ONLY ability score modifier for the sake of Graze.

’If your attack roll with this weapon misses a creature, you can deal damage to that creature equal to the ability modifier you used to make the attack roll. This damage is the same type dealt by the weapon, and the damage can be increased only by increasing the ability modifier.’

It doesn't say, ability modifier + other abilities that add bonuses.

Sacred Weapon adds a bonus, that bonus is equal to your CHA modifier. That doesn't mean you're using your CHA modifier to make the attack roll, you are using your STR modifier + the bonus from Sacred Weapon.

In your example, STR is the ability modifer used for the attack roll, then you are using Sacred Weapon to add an additional bonus. Just because that additional bonus happens to be derived from the CHA score modifier, doesn't mean that both STR and CHA are the ability modifier for the attack. STR is the ability modifier in this case, and the ONLY thing Graze considers.

At the end of the day, you are free to use whatever you wish at your table, i'm simply trying to explain that RAW and RAI - nothing increases the Graze amount over the ONE ability score modifier used by the weapon attack.