Quite a few of us have had a bug with Hancock that prevents us from completing the 'Investigate the Pickman Gallery' task that he assigns. I ended up completing 'The Big Dig', which is when you recruit Hancock as a companion, before investigating the Pickman Gallery. I investigated the Pickman Gallery some time later, went to turn it into Hancock, and was unable to. The location pip led me to Hancock, but wouldn't give me any dialog options aside from the normal companion dialog. This was probably 100 hours of game play ago. I had long since resigned myself to the fact that this quest would just forever by in my quest log.

Last night I finally got around to completing the Silver Shroud quest (awesome quest by the way), and I decided I'd take Hancock with me. I figured he'd most appreciate cleaning up the streets of Goodneighbor.

Spoilers ahead if you haven't completed the Silver Shroud yet -

Kendra is the third of your targets after starting the mission. After killing her you pick up an optional task to assassinate Kelly Tiller after finding a contract for the hit on Kendra's body. I decided to go through with the assassination. I went to the National Guard Training Yard, tracked her down, found her to be pretty uninteresting, and killed her. Right after killing her I got a notification that "Hancock hated that." Normally I'd reload a previous save when this happens, especially since I'm still trying to max out Hancock's affinity for me. However, I decided to talk to Hancock to get his 'thoughts' on the matter. Right when I clicked on him he said "When I asked you to investigate the Pickman Gallery, I never thought you'd bring me here with you!" and then he paid me the caps that he'd agree to give me a long time ago, and the 'Investigate Pickman Gallery' quest was completed. This is a bit odd. As mentioned before, I was at the National Guard Training Yard, but Hancock's dialog made it sound like we were standing in the Pickman Gallery at the time. I don't know why this action triggered the completion of that quest, but I'm glad it did. It's nice to finally have that quest out of my quest log.