Which choices below do not sound natural when you want to say someone "be hopeful" using a hopegiving sentence:
- Don't lose your hope.
- Don't be hopeless.
Why Ngram doesn't show any particualr result for them?
Ngram doesn't like these because neither is particularly idiomatic. The common expression is
Don't lose/give up hope.
This remains the same even with specific subjects, or general observations
Don't lose hope that your mother will recover from her illness.
Don't lose hope in the future.
In general, we don't say "your hope", possibly because it's not a good thing to lose, no matter who the hope belongs to. However you can use a possessive to distinguish someone's personal feelings from those of other people. Ngram has this most frequently in a religious context
Don't give up your hope in God
You can express this same feeling in the negative (with "hopeless") but it's better to soften the expression, as "don't be X" is often perceived as criticism, not encouragement.
You shouldn't feel hopeless.
There is no need to be hopeless. There is always hope.