Is this sentence, "try to relieve her by lightening her mood", correct, and if so, what is the difference here between lighten and relieve?
Yes, that sentence is correct.
Relieve: to free from anxiety, fear or pain.
Lighten: to cause to become cheery or glad.
"Lighten" is a synonym of "cheer up". And "relieve" is a synonym of "alleviate".
The difference is pretty clear since relieve does not imply to have a good mood, while lighten does.