People grow one year older every New Year's Day by eating rice cakes.

(What is the meaning of 'grow' in this sentence? What other words can we use instead of 'grow'?)

I'm not sure "grow" is the right word here. Grow: http://www.answers.com/topic/grow

I'm also not sure that eating rice cakes is instrumental in a person's reaching his/her next birthday! Are rice cakes symbolic on New Year's Day for a particular reason?

You might try this:

People become one year older on New Year's Day.

In this context, the word "grow" means to advance in age or to become older. It is used figuratively to express the idea that people age or get older with the passing of each New Year's Day.

Here are some other words that can be used instead of "grow" to convey a similar meaning:

- Age: People age one year older every New Year's Day by eating rice cakes.
- Mature: People mature one year older every New Year's Day by eating rice cakes.
- Develop: People develop one year older every New Year's Day by eating rice cakes.
- Progress: People progress one year older every New Year's Day by eating rice cakes.
- Advance: People advance one year older every New Year's Day by eating rice cakes.

Remember, using different words in a sentence can add variety and richness to your language.