Christina decides a keyword which will make her believe Hououin Kyouma's story in the next time-line. That password is:
私が今一番欲しい物はMy-Forkであり。
My-Spoonはもうもってるし。
The thing I most want at the moment is "my fork".
I already have "my spoon".
She blushes as soon as she says it. And when Kyouma repeats her this password in the following time-lines, she feels strongly ashamed (especially for the "spoon" thing) of it everytime.
What is the reason behind Christina's intense embarrassment?
According to an interview with the scenario writer, Naotaka Hayashi, featured in Dengeki Games Magazine about the heroines of anime/game that weren’t explained in the main story:
Why did Kurisu want her own fork?
Ever since Kurisu came back to Japan she couldn’t get enough of instant ramen. At first she tried using chopsticks to eat her ramen but never could get used to them, so she gave up and used a fork instead. This is why she wanted her own personal fork to eat her ramen at the lab.
This is probably the actual meaning she implies. Her comment about how Okabe shouldn't take an interest (i.e., ask more about it) in that, is probably something to hide he embarrassment of not being able to use chopsticks.
Note that she brought her own spoon for pudding, but she doesn't have a fork (for ramen).
According to the Tv Tropes page, the explanation for the "fork" and "spoon" is an inside joke from 2chan for "lover" and "friend":
Apparently "my fork" and "my spoon" was at some point 2chan slang for "lover" and "friend", respectively. No wonder Kurisu was so pissed with herself.
But no source was provided.
Though the Steins Gate wiki has a different explanation for it:
As a way for Makise Kurisu to understand that Okabe time-leaped back a few hours prior to Mayuri's death, she tells Okabe that she needs a "My Fork" to go with her "My Spoon" in the present, so that when he time-leaps back, she'll believe that Okabe came back from the future to save Mayuri.
But I still think it's just slang.
This answer will not provide you with any deeper meaning on what it would mean to own your personal spoon and/or fork, but it will tell you how Makise got her spoon and why she wanted a matching fork.
In Steins;Gate: Aishin Meizu no Babel (The story from Makise Kurisu's POV) you can see how Makise gets her personal spoon for her tenth birthday from her father and was supposed to get her personal fork from him a year after that on her eleventh birthday.
Then on her eleventh birthday, she gave her father proof that time machines should be impossible to build. He became outraged over this and instead of giving her anything, he told her he would prove her wrong and erase her from this earth.
So I guess her wanting her personal fork, would mean that she wanted to patch things up with her father and become wanted again by him. Which can be seen when Ferris lets her listen to the tape both their fathers had made 16 years earlier, when she called him after all these years.
Not much had changed though, he still hadn't been able to built the time machine and he still wanted to erase her existence in 2010.
In addition to Jon's answer:
In Japan, people usually have their personal set of dishes, which they may call my+something. e.g マイ箸 for chopsticks...
It is also trendy to prepend this "my" prefix for many personal objects that are seen as an achievement : house (マイホーム), car (マイカー)...
Combine those 2 hints for better understanding of the joke :)
