Atomica gives:

An evergreen tree (Arbutus menziesii) native to Pacific North America, having leathery, glossy leaves, white urn-shaped flowers, and orange or red edible berries. The wood has been used for making furniture and the bark for tanning.

[American Spanish, from Spanish madroño, strawberry tree.]

The next obvious question is how does madroño come to mean strawberry tree?

(Aside: in Italian, you guess rightly that madrone would mean "big mother", though maybe it would have to be femminine (madrona). That's how I first read the title of the post, too. Interestingly, the words "madrina" and "padrino" mean - not literally a little mother and father - but figuratively so: godmother and godfather!)