Update: HERE's the weapon Arya wanted Gendry to make!
In the Battle of Winterfell.
Of course, Arya has become quite the deadly assassin in the years since they last saw each other, while Gendry had gone back to being a blacksmith. But instead of constructing armor and swords for the Lannisters in King's Landing, Gendry is now arming the united armies of the North to defend the living against the dead. And so, it only makes sense that Arya would want to increase her stockpile of weapons by asking Gendry to make something for her. But just what exactly does Arya want Gendry to make?
Based on the designs of Arya's drawing, it appears that she wants Gendry to make her a dragonglass type of rocket launcher. Or, more accurately, a spear launcher. Having a dragonglass-based projectile weapon would be immensely helpful during the Battle of Winterfell. And Arya using Needle as well as her Valyrian steel dagger wouldn't be of much help, unless she was fighting in close-quarters. So, a spear launcher with arrows made of dragonglass would undoubtedly be useful for her.
From what was shown in the Sandor Clegane, aka the Hound, felt the same way when he wanted Gendry to make a dragonglass-based axe for him.
Obtaining dragonglass was a crucial plot point of Game of Thrones season 7, and now that they have as much as they need (or could carry on their ships), they - and, specifically, Gendry - will be making as many weapons as possible. Arya's personalized spear is going to be one of many. Perhaps something even more interesting will come out of all of this in the end.
Game of Thrones season 8 continues Sunday, April 21, at 9 pm EDT.