Options that increase chances of successful ranged attacks for a blind Ranger with blind fighting:
I expect a long list of options so I make this a as a Community Wiki post. I tried to make some grouping of similar effects, but if it seems bad, feel free to regroup differently.
Archery fighting style PHB 72
You gain a +2 bonus to attack rolls you make with ranged weapons.
This will require you to get a second fighting style, either by multiclassing e.g. into Fighter or by taking the "Fighting Initiate" Feat from TCE 80.
Be unseen
By the rules if you are an unseen attacker, you gain advantage on attack roles. This leads to the Fog Cloud Paradox, so your DM might have a contradicting house rule, but RAW, being unseen would cancel out the disadvantage.
How to be unseen? You have multiple options:
- Be hidden by means of a good stealth check. I believe Rangers get Pass without a Trace which can help with that. (PHB 192)
- Be invisible e.g. with the invisibility spell or with a potion of invisibility.
- Be in a heavily obscured area, e.g. by using the fog cloud spell. (PHB 183)
Spells that give advantage
As with the unseen section, having advantage would cancel out disadvantage. You may require an ally or spell scroll or something similar to cast these.
- Truestrike (PHB 284), but it comes at a great opportunity cost, so probably not worth it. (not a ranger spell)
- Zephyr Strike (XGE p171)
Once before the spell ends, you can give yourself advantage on one weapon attack roll on your turn.
- Foresight (PHB 244), but this is a 9th level spell (not a ranger spell)
- A familiar as by the "Find Familiar" (PHB 240) spell could use the help action to provide advantage. (not a ranger spell)
Put a condition on the enemy that gives the attacker advantage
- The restrained condition (PHB 292) gives advantage, it could be achieved with the ranger spell entangle (PHB 238)
- The stunned condition (PHB 292) gives advantage.
- The paralysed condition (PHB 292) gives advantage, for example with the hold person spell (not a ranger spell).
Gain Advantage another way
Besides being unseen and with spells and conditions I can only think of one more way to gain advantage, but maybe there are more:
- An ally uses the help action (PHB 192). As mentioned in the previous section, a familiar could use the help action. An animal companion could also use the help action; at 7th level a Beast Master ranger can trigger this using a bonus action and attack with their standard action. (This has the nice effect of rhyming with canon, as the ranger who taught Drizzt--Montolio DeBrouchee--was blind yet shot accurately with an owl companion indicating targets aurally.)
Spells that increase attack rolls
Some spells can increase your attack roll. This would increase your chance to hit. As with the previous section, you may require an ally or spell scroll or something similar to cast these.
- Bless (PHB 219) adds 1d4 to your attack rolls. (not a ranger spell)
- Magic Weapon (PHB 257) adds +1 to +3 bonus to attack rolls and damage rolls (ranger spell)
- Elemental Weapon (PHB 237) adds +1 to +3 bonus to attack rolls and 1d4 to 3d4 damage (ranger spell)
Features that increase attack rolls
- If you have an allied bard he can give bardic inspiration to increase the attack roll.
+x Weapon/Ammunition increase attack rolls
A +1 ranged weapon would increase your chance to hit by +1. Similarly a +1 piece of ammunition has the same effect. Those two effects stack. There are even spells and class features to make weapons +1 weapons, but that could be a question of its own.
Close to 10 feet
Since the character has blindsight out to 10 feet, getting within 10 feet would allow the character to normally attack the enemy. However don't get to close, ranged attacks have disadvantage if there is an enemy within 5 feet (PHB 195).
Use someone else's eyes (questionable)
You mentioned that the character does not want to gain sight "as she would like to role-play her character as being blind.". What about the character staying blind but looking through another creatures eyes, like with the ranger spell "Beast Sense" (PHB 217)? If you put that beast on your shoulder, it could possibly do the aiming. This however will need a lot of DM help as the original character still has the blinded condition and therefore still RAW has disadvantage.
Beast sense states:
you can use your action to see through the beast's eyes and hear what it hears, and continue to do so until you use your action to return to your normal senses.
So after you once spend an action to see through the familiars eyes on your turn.