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:
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 alternate simpler 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.