Q:

How to nudge girlfriend to find something meaningful during her summer holidays?

I've recently returned from visiting my girlfriend - we are in an long distance relationship. She started to feel very empty and depressed when I left. She cried a lot and told me she started to stress eat and feels it hard to focus on anything but the feeling of missing me.
She usually focuses on her studies to keep busy, but now summer holidays are here and there isn't anything to really keep her mind of missing me.
During my time with her, I introduced her to some computer games (to no avail) and also playing the piano (her father bought a keyboard at a flea market while I was there).
I've tried asking her to make her own mission during the holidays like learning a song on the keyboard or trying to find a game she likes. (I admit that the games I introduced were not really her cup of tea. But she did like the puzzle platformer genre.) But she tells me she is unable to think about doing such things.
Question: Is there any better approach to provoking interest in hobbies and activities other than just suggesting they try it? 

A:

Question: Is there any better approach to provoking interest in hobbies and activities other than just suggesting they try it?  

I think you are going about this the wrong way. She is lonely and wants to spend time with you. I think the best way to provoke interest in a hobby is to start a hobby that involves YOU in some way. For instance, you said she plays piano and you want to encourage her to get into it, why not start learning an instrument at the same time? Then you could play songs together. 
I've been in a LDR before, here are a couple of the things we did together:  
Watch a top 100 movie list: We would watch the films that one of us had seen alone and watch films neither of us had seen together on skype.  
Start a minecraft world together: This is nice as you get to do something constructive together but can also log in and play by yourself to help both of you out. You said she like puzzle platformer games, find some co-op ones on steam to play together.

A:

Everyone else has given great answers, but this is just a warning really.
Some people are hobby people others are not. It has taken my Mother until she is 70+ years old to realise that doing things by herself is enjoyable, this despite years and years of my sister going through the local paper showing her clubs and activities she could undertake.
It turns out she did have a hobby for all those years moaning how bored and lonely she was.
If everyone else's answers lead to nought (which they would do with my mother) I would recommend you try to help her get a summer job, so she can save up for extra visits. This is the only thing which would have worked with my Mother.
She still complained she didn't have any friends, but at least she was busy.

