In Old Norse, Ratatoskr means "drill-tooth" or "bore-tooth".

One can therefore assume that the word rat stems from a description of it's behaviour.

In Old Norse, Ratatoskr means "drill-tooth" or "bore-tooth". It is the name of a mythical creature, a squirrel that runs up and down the tree of life called Yggdrasil, acting as a messenger between two arch enemies: the great eagle and the terrestrial dragon.

According to Albert Sturtevant, "[as] far as the element Rata- is concerned, Bugge's hypothesis has no valid foundation in view of the fact that the [Old Norse] word Rata (gen. form of Rati) is used in Háv[amál] to signify the instrument which Odin employed for boring his way through the rocks in quest of the poet's mead [...]" and that "Rati must then be considered a native [Old Norse] word meaning "The Borer, Gnawer"

_Wikipedia

One can therefore assume that the word rat stems from a description of its behaviour.