Since it's a name, I would think you're free to use whichever form you like. I don't know that naming conventions necessarily have to follow grammar rules. As such, I kind of like the first option as a name, although of course I wouldn't use it that way in a grammatical sentence. Spelling conventions are often disregarded as well in naming.
Not that there's anything wrong with 'girl', but as a name you could spell it 'gurl', 'guhryl', or any particular whimsical variation you see fit, similar to personalized license plates.
Laykegyrl
Laycgurl
Leikgirl (if it's Hawaiian!)
I'm clearly overthinking this, but you get my drift (pun intended).