I've already asked a few questions on this topic (e.g. How to handle weapons having been hidden with sleight of hand?) but this one is a bit different.

In one of my campaigns I have a player playing a multiclassed Hunter ranger/Thief rogue. They put their Expertise in the Sleight of Hand and Perception skills. They're wondering if they can use the Thief's Fast Hands class feature to draw a previously hidden weapon and gain the benefit of Sneak Attack.

The idea is that if the Investigation roll of the enemy were lower than the ranger/rogue's Sleight of Hand roll, the ranger/rogue could add Sneak Attack damage to their attack, even if they don't have advantage or otherwise qualify for Sneak Attack.

Would this houserule - allowing Sneak Attack damage on an attack made with a previously hidden weapon - cause any mechanical issues or balance issues?

This is fine

What you are proposing is mechanically weaker than just giving the rogue advantage for attacking with a hidden weapon. The game encourages you to apply advantage whenever situational things favor one side of a roll more than the other, which could certainly apply to this situation. Ruling that a hidden weapon is enough to allow sneak attacks but not enough to apply advantage on the attack roll thus creates a new middle ground that is certainly not overpowered compared to your previously existing options.

Allowing Sneak Attack with a hidden weapon is probably balanced

What you are proposing is essentially a contested check, with the rogue's Sleight of Hand against the target's Investigation, that grants Sneak Attack upon success. This is actually quite similar to an existing class feature from another Rogueish Archetype: the Inquisitive's Insightful Fighting:

As a bonus action, you can make a Wisdom (Insight) check against a creature you can see that isn’t incapacitated, contested by the target’s Charisma (Deception) check. If you succeed, you can use your Sneak Attack against that target even if you don’t have advantage on the attack roll, but not if you have disadvantage on it. This benefit lasts for 1 minute or until you successfully use this feature against a different target.

You could create a similar ability for your rogue player modeled on this ability. The main differences are that Insightful Fighting requires a bonus action, uses different skills, and lasts for 1 minute. In your case, the 1-minute duration doesn't make sense. Once the hidden weapon is used in an attack, it is no longer hidden, so it only makes sense for the benefit to apply to the first attack. In return for this, you could allow the rogue to make an off-hand attack with the same bonus action they used for this ability, using the normal rules for two-weapon fighting. So, something like:

Concealed Weapon

As a bonus action, you use a free hand to draw a melee weapon from a hidden sheath and make a melee attack with it in one swift motion. Make a Dexterity (Sleight of Hand) check, contested by the target's Intelligence (Investigation) check. If you succeed, you can use your Sneak Attack for that attack even if you don’t have advantage on the attack roll, but not if you have disadvantage on it. If you are already holding one weapon when you use this ability, you must follow the rules for two-weapon fighting as normal.

Since this new ability is based on an existing 3rd-level Rogue feature and functions similarly to it, you can be fairly confident that it won't cause major balance issues. However, you are essentially adding a new class feature, so in return you should consider asking your player to sacrifice one of their existing 3rd-level class features (for the Thief archetype, that would mean trading away 2nd Story Work or Fast Hands). This trade-off is something you may need to tweak over time once you see how powerful the ability turns out to be in practice.

Based on your comments, here is an alternative version of the ability that doesn't grant the extra bonus attack or have any special interactions with two-weapon fighting:

Concealed Weapon (alternative version)

As a bonus action, you use a free hand to stealthily draw a melee weapon from a hidden sheath. The first time you attack with this weapon within the next minute, make a Dexterity (Sleight of Hand) check, contested by the target's Intelligence (Investigation) check. If you succeed, you can use your Sneak Attack for that attack even if you don’t have advantage on the attack roll, but not if you have disadvantage on it.

The typical usage of this ability would be to draw the weapon as a bonus action and then attack as normal with your action, possibly gaining Sneak Attack.

Yes, this could cause balance issues

Mechanically, the interaction you describe could give the enemy the Surprised condition, but that's where things start to get hairy.

The problem

One of the conditions for Sneak Attack to apply is if you have advantage on the attack roll. Since Surprised doesn't grant you advantage, you're faced with houseruling Surprised to say that it does grant advantage, that's a potentially dangerous precedent to set. If your whole party later ambushes someone, the enemy losing a round of combat AND the party having advantage on that first round has the potential to be very strong.

Currently, I'm not aware of a rule that allows for what you're attempting. This is getting into the realm of video game-style sneak combat, where you sneak up behind an enemy and "stealth kill" them by breaking their neck or similar. The danger you face here is if the players start to get away with stuff like this, they're going to want to try to do it all the time. D&D isn't designed to handle things like this through regular gameplay, so it becomes tough to bend the system to fit.

The other option you face is granting Sneak Attack in extra scenarios beyond those int eh rules, and this could again set a dangerous precedent that could effectively end up granting the PC a constant damage bonus.

The solution: Use a Skill Challenge

What you could instead use is an adapted version of a 4th-Edition mechanic called a Skill Challenge. This essentially boils down to the PC having to make a series of rapid-fire skill checks and getting a certain number of successes before a certain number of failures. You're already using multiple skills here, so this could translate nicely. This 1) allows for the PC to use their skills in a meaningful way while still allowing for failure; 2) doesn't potentially break the mechanics of existing 5E rules; 3) makes it challenging enough to pull off that the PC likely won't be able to spam this trick. It also lends itself nicely to cinematic descriptions of what happens - spectacular success or failure is still spectacular, after all.