After eliminating the gangsters in front of Granny Rags' house, you will receive a more serious mission from her. Speak with her again and she will ask you to poison the Bottle Street gang's illegal distillery using rat viscera from one Dr. Galvani's office. Agree to this morbid task, as you will receive a Rune in payment afterwards. This quest involves several steps. First, you will have to find Dr. Galvani's offices. This will be easy enough, since it will be marked by an objective marker should you accept the task. Head towards the marker and stay away from enemy patrols. If you stick to the streets, you should encounter two thugs patrolling outside of a captured merchant's quarters. One will go urinate in a corner, while the other will wnader off to have a smoke. Take them both out while they're distracted and rescue the merchant. He will now sell you items just like Piero would, except this merchant also carriers a few blueprints. Don't feel pressured to buy the Sokolov Elixir blueprint right away, because you can always find another in his lab in a later mission.
Now that you've saved the merchant, head off to Dr. Galvani's lab. Enter the building through a balcony looking out over the streets. Use Blink to reach it. Galvani isn't home right now, but his manor is currently patrolled by four guards and a maid. Three of the guards will be on the first floor, patrolling its various halls. One of the guards will speak with the maid before moving up to the third floor. The maid will move into the bedroom after the conversation, and then move about the house. When the guard and the maid part ways, wait a second for the guard to get most of the way up the stairs before rushing in and subduing him. Hide is unconscious body and then rush downstairs and choke out the maid. Make your way through the halls, similarly disabling the rest of the guards. Of course, if you're looking to play through the game in a chaotic manner, you can awlays kill them all, too. It won't do any lasting damage.
When you're done clearing and looting the lower floors of the compound, head upstairs to Dr. Galvani's lab, through the glass double doors. Loot the bullets and potions from around the room, and head over to the bookshelf in the back left corner. A secret book on the bookshelf will allow you to pass into a secret room. The book glows golden when you look at it, making it a not-so-secret secret. You can grab the viscera you need from the rat specimen in the center of the room. When you're done, loot his safe to add insult to injury. The combination is 287.
Now that you have the rat guts, you just need to deposit them into the Bottle Street gang's elixir distillery. This is a bit easier said than done, but with some crafty sneaking or attacking you can get the job done and get your Rune quickly. There are lots of guards at the distillery, so crouch and be careful as you approach your waypoint. If you want to avoid guards altogether, take the pipes above using Blink. They will lead you over the main yard to deposit you right at the entrance of the brewery. If you want to play things a little more chaotic, try taking cover a t the tops of the stairs leading down into the brewery and using the Rat Swarm power to summon rats at the guard's feet. The other guards will come rushing over to help as their friend is eaten alive. Toss a grenade down the stairs while they're all clustered together and watch the limbs fly.
Note that there is a Bone Charm around the entrance to the distillery. Once you're past the front yard, explore the rafters of the storeroom. Use Blink to snap over to the caskets and again to reach the rafters and the Charm.
Enter the brewery and sneak over to your objective. There won't be much to stop you at this point. Grab a few empty bottles of red potion and fill them at the distillery before using the rat carcass to poison the lot of it. That way, you'll at least get a few free health potions out of the batch before ruining it for everyone else.
Now that the distillery has been successfully poisoned, head back to Granny Rags and speak with her again to complete this little side quest and claim your reward. Granny Rags will give you the Rune and leave you to complete your main quest.