My boyfriends lawyer advised him to buy out his ex girlfriends right to claim his daughter on this years taxes. He does not understand nor do I what it means to buy out one or the other for right to claim her? Does anyone know or understand Maine's law on this or should they just try to decide on taking turns?

LOL, there is no law for this. He means unofficially. He means that she probably doesn't make as much as your boyfriend, so he pays her something to let him claim the child, and that way they both save. But the parent who has the child for more than half the year gets to claim the child.
If they were married, and he is the non-custodial parent, he also has to get a written form from her, and the form depends on the year of the divorce. But they aren't married!
Plus, she may be getting all sorts of other tax credits as a single parent.
Unless he is paying a lot of child support.
Anyway, there is no law for buying out a custodial parent's right to claim a child.

LOL, there is no law for this. He means unofficially. He means that she probably doesn't make as much as your boyfriend, so he pays her something to let him claim the child, and that way they both save. But the parent who has the child for more than half the year gets to claim the child.
If they were married, and he is the non-custodial parent, he also has to get a written form from her, and the form depends on the year of the divorce. But they aren't married!
Plus, she may be getting all sorts of other tax credits as a single parent.
Unless he is paying a lot of child support.
Anyway, there is no law for buying out a custodial parent's right to claim a child.

My Boyfriend makes less then she does, also she is trying to get him to just let her claim her every year and she would split it with him. She is constantly lying and he is trying to get 50% custody I feel like she is just trying to con him into something that will only benefit her. He pays her $55 a week in child support and they are in court because she is trying to stop him from having his daughter half the time.

The IRS counts each night the child sleeps over with whom as who the custodial parent is for tax purposes. There's several paragraphs about what a night is, about where New Year's Eve goes, about being with someone else or away at camp! If they are exactly 50 - 50 on actual nights, the parent who makes the most gets to claim the exemption.

Parents fight over this all the time and many will both claim the child, and the IRS does catch it, even if it takes 2 years, and then they contact both parents about who is going to get the right, and they sock the other parent with interest and penalties.

You boyfriend should insist on alternate years if he gets joint custody. She will say she'll split it but no one ever does when there's fighting going on.