Several of the posts have been wrong.
The rules CHANGED in 2005 (14 years ago!) and the issue is whether or not the child supported themselves, NOT who actually paid it.
For 2018 and earlier years--when you and the child both lived with your mother, either you or your mother could claim the child [unless you were a dependent too], but if you could not agree, then you (as parent) would win the tie-breaker and had the superior right to claim the child. This was true even if your mother (as grandparent) had paid everything and filed first. You can even amend 2016, 2017 and 2018 to change your return to claim the child and any other benefit you are eligible for. (Note, you probably can't claim HOH as that requires you to pay more than half the cost of keeping of the home *and* claim the child.) Your mother would be forced to amend her tax returns as well, removing the child, HOH, etc. I wouldn't recommend that.
For 2019, since you have already lived there 6 months, you will have decide if you will let your mother claim him or not.
Once you do move out and don't live with your mom for more than 6 months of the year, she'd no longer eligible. If she files electronically and claims him before you can file, then you file by mail and claim him. The IRS will straighten out the duplicate claims--you then show your lease showing that he lived with you and copies of school records showing your address.