I am also surprised that "llegar a ser" didn't come up until neizan mentioned it. I'm also surprised, but to a lesser degree, that "quedar(se)" did not also make it into one of the answers here, nor was it a part of the previously referenced document, "23 Ways to Translate Become in Spanish." So, I'll take a minute or two to share with you what I've recently learned about it myself:
With regard to "quedar(se)" and examples of its use as a verb that means "to become," a book called A New Reference Grammar of Modern Spanish devotes a couple of small sections to this verb (at least it does in its 3rd edition). I've included a bit from the book below (formatted by yours truly for emphasis/easier reading):
The relationship between "quedarse" and "quedar" (when they are used as verbs of becoming, i.e., are followed by an adjective or participle) is affected by regional considerations that make it difficult to be precise about usage. In general, it seems to us that "quedarse" as a verb of becoming is much more common in Madrid, while "quedar" has a literary or regional character.
(a) In many cases, "quedarse" implies loss, incapacity or disadvantage:
Se quedó ciego/mudo/impedido/sordo. (He became blind/dumb/disabled/deaf.)**
¡Qué delgado te has quedado! (How thin you have become!)**
Me he quedado helado esperándote. (I've gotten/become frozen waiting for you.)
(b) In a few cases it does not imply loss or disadvantage:
¿Te has quedado contento? (Are you satisifed now?)(Literally, have you become content?)
Se quedó embarazada. (She became pregnant.)***
*Some of the English translations are my own.
**The book adds that in some Spanish regions "quedar" can be used instead of "quedarse" for sentences of this type.
***Though some sentences can use "quedar" instead of "quedarse," according to the book I've referenced here, "quedar" + "embarazada" is not how you say "She became pregnant," in Madrid, but it is acceptable in other parts of Spain; in Madrid, "quedarse" would be used instead.
One last thing I should add (and the book mentions this, too) is that both "quedar" and "quedarse" have other meanings besides the ones included here. I believe both are mainly used to mean "remain" or "stay." And though you can use this verb to mean "become," it appears other ways (such as the ones listed in the document suggested by dusan) are much more common. I am unaware of any set or fixed phrases involving "become" that require the use of "quedar" or "quedarse." If anyone does know of any, please share them with us.
By the way, in an effort to help preserve the intellectual property of those who wrote the book, A New Reference Grammar of Modern Spanish -- John Butt and Carmen Benjamin -- I didn't include every example the book provides, so you may want to see if you can get your hands on a copy of it either via your library or a bookstore. It is chock full of useful information about the Spanish language, and it is written in English. Though it is rather detailed, it is easy to read and understand, and I think most students of Spanish would learn a great deal from it.