The Grimms Deutsches Wörterbuch is rather like the Oxford English Dictionary, except that it’s for German (obviously). I’m not sure as to how helpful it would be in actual language learning, but I’m sure it’d still be fun to dig around through. You can read more about the dictionary and its history here.

[OE. stréaw-, stréow, stréa-, stréuberie, f. stréaw

STRAW n.¹ + berie BERRY n.
No corresponding word is found in any other Teut. lang. The reason for the name has been variously conjectured. One explanation refers the first element to STRAW n.¹ 2, a particle of straw or chaff, a mote, describing the appearance of the achenes scattered over the surface of the strawberry; another view is that it designates the runners (cf. STRAW n.¹ 3).
The view of Kluge, that OE. stréaw- in streawberie is cogn. w. L. frgum strawberry, is not phonologically satisfactory, and is also open to objection on other grounds.]

And a bit more from Wikipedia:

The name is derived from Old English strÄ“awberiÄ¡e which is a compound of streaw meaning “straw” and berige meaning “berry”. The reason for this is unclear. It may derive from the strawlike appearance of the runners, or from an obsolete denotation of straw, meaning “chaff”, referring to the scattered appearance of the achenes.

Interestingly, in other Germanic countries there is a tradition of collecting wild strawberries by threading them on straws. In those countries people find straw-berry to be an easy word to learn considering their association with straws.

There is an alternative theory that the name derives from the Anglo-Saxon verb for “strew” (meaning to spread around) which was streabergen (Strea means “strew” and Bergen means “berry” or “fruit”) and thence to streberie, straiberie, strauberie, straubery, strauberry, and finally, “strawberry”, the word which we use today. The name might have come from the fact that the fruit and various runners appear “strewn” along the ground.

Popular etymology has it that it comes from gardeners’ practice of mulching strawberries with straw to protect the fruits from rot (a pseudoetymology that can be found in non-linguistic sources such as the Old Farmer’s Almanac 2005). However, there is no evidence that the Anglo-Saxons ever grew strawberries, and even less that they knew of this practice.

The bold is my doing.

So, apparently, my mom and wife are not alone in thinking that the name is derived from putting straw on the plants.

The explanation based on the Anglo-Saxon verb streabergen is intriguing. I’m not sure where the Wikipedians came up with it, though. As far as I can see, there’s no citation for the explanation, and the OED mentions nothing similar.