From a RAW 5e standpoint, there are no bardic instruments that are also weapons. Seeing as you, the GM, seem totally OK with this plan of action, we can assume that we just need a way to balance the item not justify its existence.
Stats - For all fighting purposes, this is a sword, the instrument does nothing to change that. I would just take a weapon such as a long sword and import that stat block entirely. You and the player can work out exactly which weapon you use as a template. Likely a martial weapon of some sort.
Cost - You mention that this is something you want the Bard to have. The simplest metric would be to take the cost of the weapon and the cost of the instrument and add them together. You mgiht consider multiplying this by 2x or 3x to account for the trouble of finding a luthier/blacksmith. You may even consider just giving the item to him due to his back story. If it gets damaged or lost you can worry about a cost then.
Usability - This one is a bit trickier. In DnD 3.X the bard would just need to take Exotic Weapon Proficiency for this item. Unfortunately exotic weapons don't exist in 5e RAW. You've got 2 options here.
1) Make up a feat - Just make the bard take a homebrew (NOT RAW) feat that lets him use his sweet Scimiguitar.
2) Use the 5e mentality and call it a martial weapon like any other. This is probably the closest to RAW you can get, but if could lead to other players feeling like the bard got a sweet toy with no negatives for free.
So now that he's got it, how does he use it?
Bards use their instruments as a spell focus that allows for bardic casting. For most bards this involves making a conscious choice to forgo holding a weapon or shield so that they can have their instrument in-hand. If you are worried about this being over powered, you could make the bard declare if he is using his scimiguitar in "instrument mode" or "sword mode" each round. This would limit his ability to make OA's and possibly other things down the line. Additionally there could be a 1:20 chance that using it as a sword would drive the instrument out of tune and make in inoperable as an instrument until the bard re-tuned it. Honestly though, I doubt that letting him simply have it both ways at all times will make it noticeably over powered.