With the coming V0.2 update, we have enough chats for Lydia to actually reach the 5 affection threshhold and finally click this button.
This was faster than expected. But Lydia is meant as the "vanilla" girl with a relatively normal dating experience, hence she has quite a lot of chat choices that can make her blush. Whether she will stay the vanilla girl is up for debate though. There will be multiple polls regarding her direction to confirm what the community wants to see from her.
The patch focuses on a more natural interaction with Lydia, where she is not distracted or in class. This way, we get to know her character better to then decide on where we want her story to go. A slight hint of this split direction is playable during her 5 affection threshhold scene.
Also don't be too worried if you can't reach this button yet: Lydia will get more and more chats, so you will be able to click this eventually, even with non-perfect chat choices.
I wonder what this button does though... Here is a little peek to help jog my memory: