Yeah but that's the quirk of language, just because one word means love in Japanese, doesn't mean that it IS the English word 'love'.
For example, in Tagalog/Filipino the English verb 'love' translates to 'mahal'. Which doesn't just mean 'love', it can also mean 'to value' or 'to consider precious'.
Now, I'm not exactly educated in the nuances of the Japanese language, but for all you know it also roughly refers to 'value'. So she might have been named 'Valuable One" or "Precious".
All names mean something. Even Anglophonic names mean something. My name "Matthew" means "Gift of God".
But getting back to "Ai". Imagine if you were a schoolteacher named "Ai". You'd be Ai-Sensei! Or "Master of Looooove!" *insert Let's Get It On as sung by Barry White*
EDIT: I also remembered that one of the quirks of the Japanese language is that 'names' can be written with different characters or words. Therefore how you choose to WRITE your name means different things. For example, "Ai" is a one-character name that can be written with different characters, including one that means 'indigo blue'. So perhaps your friend's name is "Indigo Blue".
And yes, that means Japanese puns can be VERY elaborate.