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My mother always said, "Life is like a box of chocolates."
How do you describe such phrases, i.e., a phrase/saying/quote to remember by?
Something like a memento or reminder, but for "phrases", "sayings" or "motivational quotes". "Reminder" may be a good word, but I wonder if there is something better, like "self-reminder".
The saying is categorised as a proverb on Literary Terms:
In the movie Forrest Gump, the main character frequently speaks in proverbs, and is even shown inventing a few. For example:
Life is like a box of chocolates. You never know what you’re gonna get.
This proverb means that life is full of surprises and it is impossible to predict the future.
According to CED, a proverb is
a short sentence, etc., usually known by many people, stating something commonly experienced or giving advice
I think what might fit best here is aphorism, meaning
A pithy observation that contains a general truth, such as, “if it ain't broke, don't fix it.”
It could be referred to as a proverb, as niamulbengali notes in their answer.
For the sense ("self-reminder," as you say) of something one repeats often, it could also be a mantra, defined as:
an often repeated word, formula, or phrase, often a truism:
If I hear the “less is more” mantra one more time, I'll scream.
Your question is vague and encompasses a large number of set phrases, quotations and clichés all of which have their own name/subcategory.
However: One phrase is "an old saw"
OED
old saw n. a sententious saying, a homily; a traditional proverb or maxim; a homespun aphorism;
c1395 G. Chaucer Wife of Bath's Tale 660 I sette noght an hawe Of his prouerbes nof his olde sawe.
1857 N. Amer. Rev. July 169 To clinch the argument with an old saw, is to come off with flying colors, in his own undisguised estimation.
1992 Harper's Mag. Oct. 5/1 Please spare me that old saw about a person ‘making his or her own opportunities’.
Aren't they called memorable phrases?
I also found sound bite. One of its definitions is:
A group of words, often originating in popular culture that is spontaneously popularized after widespread repeated use.
Here (Word Hippo) you can find a few synonyms of it like, catchphrase or slogan.
