Should you capitalize incomplete proper nouns?

In my book, a character is falling asleep while trying to speak, so is having trouble getting her words across.

She says "Aley and Om" (Ashley and Tom).

Should "Aley" and "Om" be capitalized?

In your case, yes, they should still be capitalized. They are still being used as names, proper nouns. Even if the pronunciation or way it is said is incorrect, it is still functioning as a proper noun and should be capitalized. It would be strange to see aley and om when being meant for names.

Names, no matter if they’re missing their first letter or are mispronounced, are capitalized. See for example this example from Harry Potter from someone with a French accent:

You remember my seester, Gabrielle? She never stops talking about ’Arry Potter