Friend gave me an old necklace of hers. What does it mean?

Welcome to Fluther!

The gift most likely means that the friendship is valuable to her. Unless she said or did something that makes you uncomfortable, it doesn’t seem worthy of second-guessing it.

It signals her interest in deepening your relationship.

Probably need more information. Could she be romantically interested in you?
Another possibility, if she suffers from depression, is there a chance that she’s suicidal and is distributing her valuable possessions?

She values you. Beyond that would only be a guess. You have to fill in the rest since you are most familiar with the situation. What do you want it to mean?

@Pied_Pfeffer Lately, I managed to visit her a few times during the last weeks, she’s been staring at and last time we separated she hugged me longer than she usually did. Obviously it could mean nothing, but I’m not most attentive guy when it comes to things like that.

@rojo I know that I value her a lot too. I’m not sure if I really like her as more than a friend. I wouldn’t be against if her gesture meant something more.

She made a huge gesture by giving you her favorite necklace. Now you really should make a decision and act upon it.
If you like her and you are visiting her more often, perhaps making a gesture back; a new necklace, bracelet, or something that means a great deal to you. This would symbolize more than a casual kindness.
If you do not have the same type of feelings, or you wish to be convinced wanting more signs, please be kind to her and graciously return her beloved jewelry.
The friendship will survive if feelings don’t jibe.
Good wishes.

Take her at her word. She values your friendship.

Welcome to Fluther.

As @Judi suggests, the benign motive could be – one hopes it is! – a deep and abiding friendship destined to last for a long time (whether romantically motivated or not). But what was the context of the gift? The demeanor of your friend as she offered it? Because it could be – one hopes not! – a symbolic divesting of earthly possessions in advance of a suicide.

@CWOTUS The context was me returning to Sweden from Germany, and she gave it to me as a reminder of. Her demeanor was how it usually was, her being very cheerful. She did though sometimes stare at me throughout my stay, and even hugged me a few times, something she rarely does with me.

In that case I would take it as a sign of pure – and high! – affection, and hope to reciprocate at some time in an equally meaningful way.

Welcome to Fluther!

It sounds like a very strong, deep friendship. I would accept such a gesture at face value.

If you find yourself having romantic feelings, and you feel like these feelings should be explored, I would discuss that with her.

My wife and I had such a friendship, and we decided to explore those romantic feelings. We agreed from the beginning that we would remain friends, no matter the result of the romance. That was over 30 years ago, and she’s still my best friend.