So I DM for a player who has a bbeg (really powerful succubus) written into his backstory that he was enthralled by. This character (Ducky) was meant to be temporary (levels 1-6) and then the player wanted to switch to his "real character" once the party either killed succubus and Ducky, or killed succubus and Freed Ducky. Either way, Ducky was gonna leave and his other character would come into play. We worked this out together, and I as the DM was fine with it under the stipulation that it may not go down exactly as he planned and that I wouldn't go out of my way to guarantee that it happened.

Whelp.... none of that matters anymore anyway because Ducky died (no chance of resurrecting) at level 3! Now they are face to face with his succubus controller and the fight is GOING to happen.

How can I customize a succubus to make it unique, but not crazy powerful?
I had originally thought to give her spells when I anticipated the party fighting her at level 6, but now that they're only level 3 (and atm pretty tapped for resources) I don't wanna buff her too much.

I'd say just leave her as-is, but plan out some potential escape routes the party could take to run away from her if they realize they're in over their heads. Come up with a non-TPK failstate if they don't manage to defeat her. Maybe they get roped into doing her dirty work for a bit until they can figure out a way out of it.

I’d agree with transmorpher, just use her off-the-shelf. A CR4 against level 3 characters should be fine. And since she’s no longer relevant to a character, there’s no real need to make her particularly interesting.

I think its worth mentioning that its 6 level 3 characters lol

maybe just bump up hp? I feel like they'll take a single target out in a round 

If you want to buff the succubus up a bit without making her too dangerous to the group, then I'd just add on some defensive aspects to ensure she doesn't drop in the first round or two. Just a few quick ideas on things that would make her tougher without making her too deadly in return.

• Increase Hit Dice / Hit Points: Agree with your through about bumping up her hit points a but, which means she can take a few more hits without being incapacitated.
• Magic Resistance: Might not matter too much with 3rd-level characters, but advantage on saving throws against spells would help ensure that she's less likely to be affected by a spell.
• Fiendish Blessing: Like the cambion ability, maybe this succubus gains a bonus to her armor class based on her Charisma bonus. Adding a +5 bonus might make her a little too hard to hit, but a higher AC in general would deflect some of the attacks against her.

None of the above increase her offensive capabilities outside of potentially lasting longer in battle, but it would make her a little stronger (and a bit more unique) in the eyes of the party. Agree with the others about not creating something overly detailed if she's probably not going to make an appearance beyond this encounter. Plus, if you really don't want to put much effort into it, just say she has 66 (+X) HP. If party does more than 66 point of damage in the first round, then she has just enough HP (i.e., X amount) to get her into round 2 before she's defeated.

Okay, I see your problem now... I thought you were just worried about killing your party and didn't realize you were also concerned that the Succubus would be defeated too easily. In that case, I'd recommend including some more fiends. Either another Succubus or pick some weaker fiends to assist her... maybe toss in some imps or a Hellhound or two. Give your players a meat wall they're going to have to get past in order to reach her so she'll have the chance to charm someone or use her other nifty abilities.