I can see why you are confused. You've asked what seems like a simple question, and you've received answers from three native English speakers who all disagree about the answer. Please let me try to explain.
There are two approaches to grammar: prescriptive and descriptive. Here's what the BBC says:
"A prescriptive grammar is a set of rules about language based on how people think language should be used. In a prescriptive grammar there is right and wrong language. It can be compared with a descriptive grammar, which is a set of rules based on how language is actually used."
Kris, a community tutor, has taken the first approach. She's studied some grammar rules and has learnt that 'me' should be for the object of a sentence. This is why she says that 'me' is incorrect, because it appears to break this rule. She's basing her answer on what she knows about formal grammar.
The other two members are basing their answers on how native speakers actually use the language in the real world. They hear and use 'me' in this context every day, so they say that this is right; they don't hear or use 'I' used in this way, so they say that this is wrong.
So are there any mistakes? No, there aren't. They are all OK, but in different ways. This is what I think:
1. She comes later than me ---> I think this is fine. We all say this.
2. She comes later than I ---> This is technically correct according to prescriptive rules of grammar ( based on Latin!) but sounds very strange. Nobody says this.
3. She comes later than I do ---> Fine. This is a useful compromise if you want to avoid the dilemma about which of the two other options to use.
My advice would be to use the 'me' option. If it's a more formal written context, you could get round the problem by using the 'I do' option. Forget about the 'I' option.