When Steve Rogers visits Peggy Carter, she has a family photo on her nightstand and says that she married someone after Cap disappeared. Who did she marry?
TL;DR: Given what little we know, the most likely candidate is SSR agent Daniel Sousa.
The "Agent Carter" series was cancelled after two seasons. During its run, however, Peggy had often found herself attracted to fellow SSR agent Daniel Sousa, one of the few men to treat her as an equal and not a secretary. In the final moments of the last episode, S02E10 "Hollywood Ending", Carter and Sousa share a passionate kiss and begin laughing.
Sousa was engaged to his girlfriend Violet at the time, but previous scenes had made it seem like he was doing so only because Peggy hadn't returned his interest. Assuming that he began a relationship with Peggy instead, it is likely Sousa that she married. This is supported by the fact that her mystery husband was saved by Captain America, and Sousa once told her that Captain America was present at the battle where his leg was wounded.
Specifically, the relevant quote from Peggy in the Smithsonian interview was:
A blizzard had trapped half our battalion behind the German line. Steve...Captain Rogers, he fought his way through a Hydra blockade that had pinned down our allies for months. He saved over a thousand men, including the man who would become my husband, as it turned out. Even after he died, Steve was still changing my life.
Had we gotten a third season of the show, it's entirely possible that we'd find out Captain America is the one who saved Sousa after he was shot (he didn't know who had found him - merely that he woke up in a hospital with his leg gone). Some assumptions/speculation is needed here, but what little we know of Sousa and the mystery husband do fit with a little nudging.
Note: Avengers: Endgame writer Stephen McFeely recently stated that (spoilers for Endgame)
Steve Rogers was always intended to be the mystery husband.
Personally, I don't see how this lines up with the evidence from previous movies & shows, but felt I should add it since it IS relevant to the original question.
This is not addressed fully within the confines of The Winter Soldier, but the writers of the film, Christopher Markus and Stephen McFeely have recently specifically addressed this and the theories that ensued after Avengers Endgame.
Perhaps the most hotly debated Endgame plot point is the fate of Captain America, who travels back in time to live out his life with Peggy Carter (Hayley Atwell). 2014's Captain America: The Winter Soldier established that Peggy had a husband and two kids, and some fans have theorized that Steve Rogers secretly was this husband all along. Others have stated that Steve created an alternate timeline when he went back to live with Peggy, and somehow got back to the mainline MCU as an old man.
Markus and McFeely accept that different people will have different viewpoints on this topic. But, in their minds, Steve was Peggy's husband all along.
"It was always our intention that he was the father of those two children. But again, there are time travel loopholes for that," said McFeely.
Added Markus: "It does introduce the idea that there are two children who have somewhat super soldier DNA."
However, this does possibly contradict what the Russos have themselves said, saying that it would have been a branched/alternate reality that Steve had to have lived in if he went back to live with Peggy.
“If Cap were to go back into the past and live there, he would create a branched reality,” Joe Russo told Entertainment Weekly. “The question then becomes, how is he back in this reality to give the shield away?”
So, which is correct? For now it sort of depends if you give more weight to the directors or the writers.
The husband is not named.
People have asked who Peggy Carter's husband is. The movie did not say, and if you look carefully, you'll notice her family pictures are conspicuously missing a husband. As mentioned above, there's talk of an Agent Carter TV series, and it's likely the producers don't want to give away who her love interest will be. If I had to guess, I would think they would pair her up with Jeff Mace, the third Captain America (there have been many). He was Cap for the late 40s and married the female secret agent from Cap's origin.
You can read more about the somewhat tortured backstory around that agent and marriage from the comics here.
I don't think it's going to be Gabriel Jones. My guess is the British guy from the Howling Commandoes. Just guessing though. The new guy on S.H.I.E.L.D. is Gabriel's grandson, and I think it would have come out if Peggy were his Grandmother. Also there were only two soldiers marked deceased when Cap was reviewing stuff on the Howling Commandoes and I'm betting one of those was her husband.
At no time did it say he wasn't a howling commando and the best bet is Falsworth/Union Jack (the English guy with the Beret) while Gabe Jones IS a possibility I would be more inclined to think that Trip would be her grandson through a mixed marriage... I'd be interested to see if they went the older Luke Cage route and Jessica Jones being her daughter if it 70's set and as I expect Daredevil is more modern... after all they can't use the Spidey origin for her
You'll soon see when they announce casting for Agent Carter, if Falsworth is in it in any form then it's likely him.