It's on the weaker side
This item is most similar to the published Javelin of Lightning (Uncommon) , which has a similar area of effect thrown attack. Let's examine their key differences:
- The Javelin is always a magic weapon, regardless of how it is used.
- The Javelin's special attack damage is significantly higher at 4d6 with the same DC (DC13) and it deals half damage on a successful save.
- The Javelin's special attack's range is significantly larger with a 120ft line.
- The Javelin's special attack always activates, regardless of whether it hits.
- The target of the Javelin's special attack also suffers bonus damage (4d6) if they are hit.
- The Javelin's special attack can be used only once per day.
- The Javelin's special attack is activated at no cost (it can be applied to any ranged attack made with the Javelin).
- The Javelin does not (!!) require attunement.
As you can see, the Handaxe of the Storm is weaker in almost every conceivable way. The sole exceptions to this are number of uses and the fact that the handaxe can be used for a third bonus action attack without requiring Two Weapon Fighting.
As such, there is significant room for buffing up this weapon. As is, I'd be concerned about your Barbarian feeling disappointed or frustrated with both their item allotment and attunement slots being occupied by such a weak weapon and I'd expect them to ditch it for another +1 weapon at the first opportunity.
My suggestions
Action Economy
The fact that this weapon requires a bonus action to use is rather unusual and functionally only serves to complicate its usage. Consider the fact that, as a weapon with the light tag, your Barbarian is already able to attack with their bonus action, throwing their weapon or otherwise, by using Two Weapon Fighting (assuming they use it in tandem with their other +1 handaxe).
As such, it should be activated at no cost using a command word, same as the Javelin of Lightning. They can still use it in the same way thanks to TWF, but they also have the flexibility to use it via the normal Attack action on turns where they activate Rage.
To magic or not to magic?
Similarly, it should always count as a magic weapon, because your player will figure out that there's almost no functional difference. As is, your player should be attacking twice with their +1 handaxe and once by throwing the Handaxe of the Storm, using their bonus action for the third attack (using either Two Weapon Fighting or the special attack) for wholly magic damage on every attack.
The only thing that's added by making it magic only on thrown attacks is the annoyance of dealing with disadvantage if there's an enemy nearby, which they can negate with Reckless Attack.
As is, requiring attunement for a mundane weapon is simply too high a cost to pay.
Damage, Area of Effect
This is the place where you have the most room to play with. As is, the fact that it can be used 3 times per day makes up for the lower damage and reduced area of effect. As it stands, this 3*1d6+1 sums out to just half a point damage less than 4d6 so this seems to be well in line for the Uncommon tier of items, ignoring the other differences in damage (half-damage on save, range, bonus on initial target, etc.).
The only things I'd change would making the initial target eligible for bonus damage and making it deal half damage on a failed save. Additionally, I'd recommend either making the special effect always activate or increase its damage to compensate.
Final Thoughts
It's a neat concept and any player would be happy to see their DM creating a special weapon for them like this. I would just focus on cleaning up the stats and verbiage to better fit with the precedents set in the DMG (with the Javelin of Lightning) and you'll have a cool item and a happy player.