Hello, I rechecked it a lot of times, and it's really 'Me eyes', but not 'My eyes', it's from Frozen, an Irishman, as it's written in the script cannot wait to see the Queen and the Princess, and here he says:

IRISHMAN: Oh, me sore eyes can’t wait to see the Queen and the Princess. I bet they’re absolutely lovely.

So, perhaps it's old-fashioned way of expressing your expectation.

I have never seen such a usage, and don't know how to look it up or google this, then I have only you to make me sure or explain if I got it wrong

 

It's just a non-standard variant possessive meaning "my."

 

It's the written form of the Irish accent he uses. It is informal dialect.

[Cross-posted]

 

And there are previous threads here and here.