Learn Icelandic: Veðurblaðran fauk eitthvert þannig að við misstum merkið. - The weather balloon blew somewhere so that we lost the signal.

Eitthvert is a directional adverb meaning "to somewhere." Because the balloon is actively moving to an unknown destination, you need this "motion towards" form. If the balloon were just floating stationary in an unknown location, you would use the static form einhvers staðar (at/in somewhere).

Why is fauk used for "blew" here instead of a form of blása?

Fauk is the past tense of the strong verb fjúka, which specifically means "to be blown away" by the wind (used for things like balloons, hats, or leaves). The verb blása is active and is used for the source creating the wind, like a storm blowing or a person blowing on hot coffee.

What is the function of þannig að here, and could I just use svo?

Þannig að literally translates to "thus that" or "in such a way that," and it acts as a conjunction introducing a direct consequence (so that). You could absolutely use svo or svo að here instead, but þannig að is incredibly common in spoken and written Icelandic for expressing a result.

Why is merkið in the accusative case in this sentence?

The verb missa (to lose) governs the accusative case for the object being lost. Because merki (signal) is a neuter noun, its accusative definite form (merkið) looks identical to its nominative form, but its grammatical role here is strictly accusative.