I do want Tina in the game, however she needs to be added in a specific way to make sure she's balanced. Honestly... right now with your idea it sounds like Tommy would be picked more over her, purely because Tommy seems a lot more useful with his maxed out stats.
I don't believe her ability should be random. I don't mind having things that can she can trigger around the map. But just having it trigger randomly makes it pretty useless. I was thinking maybe she could pick up weapons with it and throw them at Jason. Stun chance and damage still affecting the weapons when they hit Jason, but allowing her to use her power to attack him from a distance. There would be a hefty cooldown on the power of course, as you mentioned later in your post. Her not being able to pick up weapons... well that's a problem, especially if we're going for your original idea for her power. If it has a long cooldown, then she's going to be almost powerless when her ability isn't active. Doesn't sound much like a hero character huh? So as we were discussing on the Discord, she should have low luck so her stun chance is low. (Although I'd say her ability has a base chance of 5 luck to stun, purely to make her ability a bit more useful) Her strength should also be low. (Again with her ability at a 5 for the same reason as before) But she can pick up weapons to use when she cannot use her ability, since it's on cooldown.
"Jason cannot grab Tina". That's a big no from me. She should be able to be grabbed at any point, but she can save herself if her ability is active and a weapon is lying nearby. (Like pick it up with her ability and then throw it at Jason to break her free) So Jason could still get cool kills on Tina, because we all know being slashed to death isn't exactly fun. As for saving other counselors, I don't think it should be random and should follow my idea from before. The cooldown on the ability, it's fine as a long cooldown, I agree with that.
I don't know about your way of killing Jason. I mean the part about having her father do it, sure. But finding her mother's body? Doesn't make much sense to me. I was more thinking that Tina could lure Jason to a pier or dock and then perform a specific stun on him that acts like the sweater stun, and if someone hits him (To make sure she can't do it alone, I think that if she could that would be too powerful) then Tina's father can come up and drag him down into the lake like in the films. Obviously this would need to be done after Jason's mask was off. As for taking off the mask, I feel that Tina could cause damage to Jason using her ability, since damage is the stat that gets the mask off. Therefore, if she does enough damage to him (With 5 strength with her ability mind you) his mask snaps off like in the film.
These are just my thoughts on your idea and my own suggestions as to how to add her.