resolving things with her mum is definitely important to alex and an unfinished character beat, but I don't think it's her Unfinished Business. while there is a good bit of variance, the usual way they pitch unfinished business is that it's something unresolved specifically about the circumstances of a person's death, not their life. and though cutler was the one to actually kill her, he never shied away from owning up to it, and he got crisped and kebabed right in front of her and nothing happened. hal, on the other hand, denies he has any responsibility for her death from the moment she appears to him as a ghost, and getting him to admit a share of culpability is alex's main focus for the first few days of her afterlife.
throughout season five she is consistently the one reading him to rights when he screws something up. she gets on him for recruiting crumb in secret and lying to lady mary for centuries and feeding without asking for help, but Also she consistently doesn't have it in her to go the full nine yards. she and tom stop him from killing himself, she stops lady mary from killing him, she bats the potentially lethal werewolf blood out of his hands before he can drink it, she goes back on her word that she'd stake him if she ever saw him again, and she stops tom from killing him despite knowing that it's a mistake as she does it. this is a trait of hers that we see highlighted several times as something she can't bring herself to do despite knowing it's the right thing, not entirely unlike how annie couldn't let harm come to eve. annie has her whole episode about learning that unfinished business isn't always something good or righteous, and when hal tells alex that unfinished business is usually something positive, she says she doesn't think that's the case for her.
so, the hypothesis: alex's unfinished business is properly holding hal to account, the way to seal the deal on that would be by killing him, and doing the deed would represent a major character development from her established traits. I also think that would be more interesting than having tom be the one to kill him because that would basically just be a redux of season three.