A bonus by any other name
My advice would be to grant your player some kind of stat increasing bonus, since it is the simplest way to level the playing field numerically. You said you aren't a fan of just giving her a stat boosting item so I'll try to give you a few other options to consider.
Give her a stat boosting item
Now, bear with me here, because at first blush this probably sounds just like that thing you don't want to do. But I would argue that when done right stat boosting gear can be a fun and flavorful way to pump up a player's abilities. The key is to do it in a way that makes sense for the story and that interests your player.
The first thing I would do is to find an item that is obviously perfect for your player. In this case I would suggest something like the Belt of Perfect Body, which grants bonuses to Str, Dex and Con.
The next thing I would do is to make it so that the item is thematically suited for your player. A Belt of Perfect Body would be useful to both your Gnome Rogue and Halfling Barbarian players, so if you are explicitly trying to get the Halfling better stats you need a way to make sure the Belt is obviously for her. There are a couple ways to accomplish this.
The first way would be to design your item to be strictly better for your Barbarian than it would be for your Rogue. You could do something like
This item grants an additional +2 Bonus to Str, Dex and Con while the wearer is Raging. Essentially what you are saying is "This item is made for Barbarians!" Technically the Rogue could still use it but it would obviously be more effective on the Barbarian. I would avoid more obvious limitations, like racial restrictions, since they tend to feel like you are playing favorites.
The second way to make sure items go to who you intend to give them to is to use the story itself as a delivery method. Why does your Halfling Barbarian get this useful item instead of the Rogue? Because that belt is a family heirloom and it would be shameful to let anyone else wear it. This method takes a bit more planning, since you need to come up with a plausible reason in-game for why a particular character gets a particular item. But when done right I think that it makes for much more player investment. If you say that the player's new weapon was passed down to them from their dead father, and someone tries to steal/sunder/mock that weapon, I guarantee your player will take that action personally and react much more strongly than they otherwise might.
Learn your lessons
If you are dead set against using items to power up players another option is one-time, permanent stat boosts. These can be a bit game-breaking, which is why they are so rare in the normal game. But there is no reason why you can't add a few into your campaign to get power levels where you want them. The most important thing to remember when going this route is that where these bonuses comes from has to make sense in terms of the story. If a bright light shoots from the heavens and gives a character a stat bonus for no reason, it is going to come off as cheesy and lazy.
One way to handle this is to have your Barbarian run into a grizzled veteran who offers to teach them a few tricks. The nice part about this is that you aren't limited to just stat boosts this way. The character can be taught to read an opponent's body, giving a bonus to hit. Or where the most vulnerable places on the body are, giving a bonus to damage. This way you can tailor what bonuses you give out to the character's play style without risking pushing them to the other end of the power spectrum.
It doesn't count unless you earn it
Regardless of how you handle powering up players you should make sure that you aren't just handing them things on a silver platter. If your players don't feel challenged then it doesn't matter how awesome a reward they get, they won't feel like they have earned it. If you make your party overcome some sort of challenge first, they are going to be more invested in the rewards.
Want to give a character a family heirloom? Make them clear out all of the Ghouls that have made the family tomb into a nest first.
Want a character to learn from an ancient master? They have to prove themselves worthy first.
Not every challenge has to be a fight, or even physical. But every reward should be earned, and the bigger the reward the harder the challenge to get it should be.