Ofukuro

• There are some ways of calling your mother in Japan.
• There are some ways of calling your mother in Japan. There are many different ways that you can address your mother in Japan.

  Just a slightly clearer way to say this :)

• The most popular expression is "okaasan" or "kaasan."
• This sentence is perfect! No correction needed!

• Recently, the other expression "mama" has also become gradually popular.
• This sentence is perfect! No correction needed!

• In many Japanese home, we have our children say "mama" and "papa" instead of "okaasan" and "otousan" because of the ease of pronunciation, and they seems to become household words.
• In many Japanese homes, we have our children say "mama" and "papa" instead of "okaasan" and "otousan" because of the ease of pronunciation, and they seems to become household words.

  Very interesting! As a general note, "... because of the ease of pronunciation" is fine, however you will more commonly see "... because it is easier for them to pronounce". :)

• On the other hand, we (especially men) sometimes call our mothers "ofukuro."
• This sentence is perfect! No correction needed!

• There are some theories of the etymology:
• This sentence is perfect! No correction needed!

• ・In the past, mothers managed property of their homes by putting them into bags ("fukuro" means "a bag").
• ・In the past, mothers managed the property of their homes by putting them into bags ("fukuro" means "a bag").

• ・We grow up in our mother's bosom ("bosom" is called "futokoro" in Japan).
• This sentence is perfect! No correction needed!

• I think that the saying "ofukuro" is cool, but I missed the timing of switching to "futokoro" from "okaasan."
• I think that the saying "ofukuro" is cool, but I missed the transition from "futokoro" to "okaasan". (timing of switching to "futokoro" from "okaasan.")

  I'm not sure exactly what you intended to say, but hopefully I got close. It seemed like you were trying to say that one term ("futokoro" or "okaasan") came before the other, but I could be wrong :)

Great work! Very interesting that mama and papa are becoming more popular! The part about the expression "ofukoro" is also very interesting, thanks for sharing the theories surrounding it! Very clearly written, except for the two parts that I commented on, but they were very close! :)

• I think that the saying "ofukuro" is cool, but I missed the timing of switching to "futokoro" from "okaasan."
• I think that the saying "ofukuro" is cool, but I missed when people switched from "okaasan" to "futokoro."

  I think you were trying to say that you didn't notice the transition from calling mothers "okaasan" to calling her "ofukuro." Let me know if that's what you meant. :)

• I think that the saying "ofukuro" is cool, but I missed the timing of switching to "futokoro" from "okaasan."
• I think that the saying "ofukuro" is cool, but when I got older, I never switched to "futokoro" from "okaasan."

  Based on your comment, this should be the correct way to say what you wanted to. :)

• Ofukuro
• This sentence is perfect! No correction needed!

• There are some ways of calling your mother in Japan.
• There are many ways of calling your mother in Japan/Japanese.

  日本では、母親の呼び方が何通りかあります。
  何通り = many ways
  I would say "Japanese" since you are talking about the language and not the country. Although, it is not wrong to say "Japan."

• The most popular expression is "okaasan" or "kaasan."
• The most popular expressions are "okaasan" and "kaasan."

  最も多いのは、「お母さん」「母さん」です。
  
  Since there is more than one noun, we have to use the plural form of "expression" (=> expressions) and "is" (=> are).
  
  We use "and" because we are listing nouns.

• Recently, the other expression "mama" has also become gradually popular.
• Recently, another expression, "mama," has also become gradually popular.

  最近では、「ママ」という呼び方も増えてきました。
  
  Since the expression you are about to talk about has not yet been mentioned, we treat the noun as a non-specific noun. Non-specific nouns use the indefinite article ("a"). "another" is the indefinite version of "the other."

• In many Japanese home, we have our children say "mama" and "papa" instead of "okaasan" and "otousan" because of the ease of pronunciation, and they seems to become household words.
• In many Japanese homes, we have our children say "mama" and "papa" instead of "okaasan" and "otousan" because of the ease of pronunciation, and they have become household words.

  子供の頃、発音のしやすさから「お母さん」「お父さん」の代わりに「ママ」「パパ」と呼ばせる家庭が増え、それが定着したものと思われます。

• On the other hand, we (especially men) sometimes call our mothers "ofukuro."
• This sentence is perfect! No correction needed!

• There are some theories of the etymology:
• There are some theories of the/this etymology:

  語源としては、
  
  I would say "this" since you are about to describe what you just mentioned (the origin of the name, "ofukuro")

• ・In the past, mothers managed property of their homes by putting them into bags ("fukuro" means "a bag").
• ・In the past, mothers would keep her family's valuables in a bag and manage them ("fukuro" means "a bag").

  かつて母親は家庭の財産を袋に入れて管理していたからという説や、
  
  I had to reword this sentence.
  
  家庭の財産 = family property; family valuables. "Property" is okay, but in "property" is usually used for large things (land, a house, vehicles, etc.); "valuables" is better because it is often used to describe small things (like money, jewelry, etc.)

• ・We grow up in our mother's bosom ("bosom" is called "futokoro" in Japan).
• This sentence is perfect! No correction needed!

• I think that the saying "ofukuro" is cool, but I missed the timing of switching to "futokoro" from "okaasan."
• This sentence is perfect! No correction needed!

Interesting origin of this word. I thought it was going to be related to owls.