My friend has a child with someone she is no longer dating. They were never married, however paternity has been established through the child support system. The child's father occasionally pays child support but not all the time, and rarely visits the child.
If my friend was to take her 3-yr old to the child's fathers house and leave the child with him and leave for two months and come back, is it child abandonment?
She's writing a letter with all her daughter's allergies and doctors numbers and everything on it, and her house phone number so the dad can contact her, but he's not going to be aware that's she laving the baby with him until he sees her drive away...

It is not since she is arranging for the care of the child. They should do the agreement in writing, to prove it was agreed to, and state why even.

The father could if they did not, in about a month lie, and try to get a change in custody based on it, but if it is shown that it is a temp issue, there should be no issue, he is merely getting an extended visit with the child