Over the weekened one of my players stumped me when she made the claim that Dispel Magic wouldn't work on Tiny Hut due to these lines:

Creatures and objects within the dome when you cast this spell can
move through it freely. All other creatures and objects are barred
from passing through it. Spells and other magical effects can't extend
through the dome or be cast through it.

I folded and just went along with the players, but my interpretation was that Tiny Hut has the ability to block magical spells but is limited to those which are trying to go through it -- not spells cast at the barrier itself, right?

You are correct, dispel magic will work on the hut.

Spells and other magical effects can't extend through the dome or be cast through it.

Casting dispel magic would be casting a spell at the hut, not through it (eg to a target inside or on the other side of it). A caster standing on the outside of the barrier casting a dispel magic at it, would be targeting the barrier and would not require the spell being cast to travel through the barrier itself.

Thus, there is no reason why dispel magic would not work on it.

Tiny Hut blocks spells on creatures or objects inside the hut; but does not block spells on itself

To give a concrete example: If a spellcaster were to cast Dispel Magic on a creature that they knew was inside the dome, the spell would fail, regardless of whether they were physically capable of seeing their target or not (a circumstance that is contingent on, among other things, whether the inside were darkened or not, per the spell description).

But if that same spellcaster were to target the hut itself, the hut would be eliminated. There aren't rules in 5th edition that adjudicate the idea that Dispel Magic—or any other kind of abjuration magic—is required to "pass through" its target as part of the mechanism of affecting it; there's only the implied requirement that a spell must reach a target, passing through any obstacles between the caster and its target. As a result, a spellcaster casting such a spell would only have to make the spell "pass through" the hut if the spell specifically says that's what it does, which Dispel Magic does not.

Choose one creature, object, or magical effect within range. Any spell of 3rd level or lower on the target ends. For each spell of 4th level or higher on the target, make an ability check using your spellcasting ability. The DC equals 10 + the spell's level. On a successful check, the spell ends.

—Dispel Magic, Player's Handbook, pg. 234