Bucky got out of Azzano late October-Early November, so if we factor in however long it took Steve to get the Commandos into being, plus their time in the infirmary for however long, it's possible that the group was on a HYDRA run during Bucky's birthday in 1944. Bucky also fell in early 1945, estimated around late january, early february, which at this point, was probably a couple weeks or so from his birthday on March 10.
After all, Steve grew up in the Great Depression, he wasn't as well-off as Bucky's family, so it had to suck when he was invited to Bucky's birthday parties and all of Bucky's other friends brought gifts from department stores while Steve's gift is something he handmade or from the cheap store around the corner, stuffed inside a brown paper bag that was decorated with some fancy drawings on to look nice because he couldn't afford ribbon. We all know Bucky treated whatever was in that bag like gold every time no matter what, and he prized everything in his room and never threw away a single thing Steve ever gifted him, but hey, Steve knew. So with his first real paycheck, his first army check from the government, you think he got excited around Christmas when he realized that not only could he get something nice for Bucky for Christmas, but he could also get him something special for his birthday?
He probably spent sleepless nights wondering what he could get that would really make Bucky get excited, he probably stopped outside store windows in London even if war was going on and people were struggling to find nice things, he probably still looked around on shelves and through windows to find the perfect birthday present and fought Phillips to get them to take at least a couple days off around that time because goodness knows that for all of Steve's righteousness, there was no way in hell he was letting any birthday slip past, not even the Commandos, but even if they couldn't celebrate in the Whip and Fiddle, Steve was determined to take a good bottle of whiskey on any mission and that carefully, specially wrapped present with him so even in the dead of night, he could share his first birthday drink with his best friend.
After all, it was the first time he could really show Bucky just how special he was to Steve. After all the presents of Bucky getting Steve colored pencils and trips to places and new shoes and everything, Steve finally gets the chance to give him something special in return. After all this time, Steve gets to give back and he wants to, so desperately.
So what do you think it was? The package Steve didn't get to give him, the paper-wrapped box with a twine ribbon on top, so carefully, perfectly folded so every crease was almost department-store quality, the one that sat alone in Steve's room underneath his cot when he dove into the Atlantic? Do you think the army took it? Did Peggy? Do you think the government took it to keep from the public the fact that the Great Captain America approved of materialism or that he cared that much about his "best friend" to go this far? Do you think the museum dug into it and ripped it open gleefully to put it under a glass display case, just to watch when a freshly thawed Steve broke the glass without even blinking and tucked it back away where no one would ever see it again, eyes glassy and his face pale?
What do you think it was?