Why was Ned never betrothed to anyone in his youth? + Who would have been a potential match for him, had Rickard and Brandon lived as they expected to?
Well, Ned was only about 18 as of the Tourney at Harrenhal, and men in Westeros often marry much later than girls. (Who may marry after flowering, but far more typically near adulthood at 16.) With Brandon and Lyanna betrothed south, Rickard was probably holding out for something else with Ned… possibly another southern match, if the Southron Ambitions theory is correct. Or possibly Rickard was considering a northern match for Ned, to mollify his bannermen who would have been expecting a Stark for one of their children. (Such as Lord Rodrick Ryswell, who pretty much threw his daughter Barbrey at Brandon, and tried for Ned too.) A second son isn’t as prestigious as the next lord or only daughter, but any Stark is nothing to turn up your nose at. Either way, most likely Rickard was judging all the options, waiting for feedback from his bannermen after Brandon and Lyanna’s weddings, waiting to see how things were going with Mad King Aerys, before deciding to do anything definite with Ned.
So, of southern ladies, there’s lots of possibilities. To match the Stark connections to the Riverlands and Stormlands, a Vale lady could be an option, maybe a Royce or a distant cousin via Jocelyn Stark (Waynwood, Corbray, or Templeton), though Ned’s fostering with Jon Arryn was probably alliance enough. For Dorne, @goodqueenaly has a great post about the pros/cons re a marriage to Ashara Dayne re Southron Ambitions, which would apply to other Dornish bannermen too. The Westerlands were being handled by the Tullys at the time, and Tywin would have said hell no for Cersei anyway, and any other house there wouldn’t have been much use. And if SA was trying to reach out to the Reach too, there’s Janna or Mina Tyrell, or maybe a Hightower girl (though not Alerie, she was already married).
But if Rickard was planning on keeping Ned in the North, for balance and bannermen-sweetening (which Barbrey thinks unlikely, but y’know she is slightly paranoid), there’s quite a few northern ladies to list. Though first I should note that Rickard might have considered Barbrey Ryswell damaged goods because of Brandon, and if so it’s possible there weren’t any other sufficiently highborn northern ladies who’d flowered yet, which could also be why there hadn’t been any betrothals made. For example, Sybelle Locke (now Glover) has rather young children now, so it’s probable she would have been way too young at the time. Berena Hornwood’s eldest child was born c.285, so she’s probably a little closer to the right age, but perhaps was still too young for marriage or betrothal back then. Donella Manderly would certainly have been old enough (if not too old), but was probably already married or betrothed to Halys Hornwood. Jonelle Cerwyn would actually have been just the right age (14, in 281) for betrothal at least, though she’s “plump and homely” so maybe Rickard thought she wasn’t good enough. And Bethany Ryswell was also married already (Domeric Bolton was born c.281 or before that), as was the unnamed Glover girl (married to Jorah Mormont). After them… um… Leona Woolfield (now Manderly)? Nope, the Woolfields are likely sworn to the Manderlys, and therefore not a house high enough for Ned, or for Rickard’s plans rather. Jyana Reed (whatever her maiden name might be) would also likely be not quite highborn enough as she’s a crannogwoman from a house sworn to the Reeds. And, well, that’s it for named Northern ladies – there would have been others, I’m sure, in the mountain clans and in the rest of the North, but I don’t know their names so I can’t judge their matchability.
Anyway, there’s a nice list of possibilities for Ned for you. Hope that helps!