Does doing research at a highly prestigious university like an Ivy League as an out of state student help with admission to that school? Would a supplemental LOR from the professor help?
Research experience may help with admission at some schools, but not at others. It depends on the length of the experience, the level of responsibility you’ve had and what kind of impact and impression you make with the PI of the lab.
As an OOS high school student, at best you’d be in the lab for a few weeks over the summer. It will be difficult to make a strong positive impression in such a short length of time., especially since many lab have layers of workers–permanent, full time paid assistants, post-docs, grad students and undergrad students–who are all ahead of you in experience and level of lab responsibilities.