The common theory is that the origin of the word “strawberry” is based on the practice growing the plants with a layer of straw mulch to help retain moisture, deter pests, and keep the berries cleaner for picking. There is also a thought that they were simply packaged in straw when being brought to market.
There is one problem with this theory which in this article from Snopes sums up very nicely:
The word ‘strawberry’ has been part of the English language for at least a thousand years, well before strawberries were cultivated as garden or farm edibles. They grew in the wild of their own accord, their berries gathered by folks who happened across strawberry patches. Therefore, although the word’s precise origin cannot be determined, we can certainly throw out any etymology dependent upon the plant’s being bedded in straw to protect it from the cold or keep its berries clean. Even if such practices had been used, they would have begun long after the fruit had been given its current name.
So where did the name ‘strawberry’ come from then? There are a couple more theories based on the meaning of the word “straw”
One explanation hinges on a particular meaning of "straw" a particle of straw, chaff, or mote describing the appearance of the achenes (tiny yellow seeds) scattered over the surface of the berry. Another view says it comes from "stray" or "strew," designating the spread of the plant’s runners (slender tendrils), in that in an unchecked field of strawberries the plants appear to have been strewn across the ground with their runners straying everywhere.
In the end I guess we really do not know for sure where they got their name, so any other theories out there?