Recently, I have completed my PhD in Biomedical Engineering from a 25 - 30th ranked USA university and got a Post-doc offer from a 9-12th ranked USA University. To keep me, my PhD advisor offered me a non-tenured tracked Research assistant professor (RSP) position with 1.5 times of offered Post-doc salary. I will be funded through my advisor grant and need to conduct research related to the funded project. However, I will also have the independence to pursue my research interest and apply for external funding. I have a very good relationship with my PhD advisor and good publications in my PhD. Eventually, I want to get a tenure-tracked faculty position. What is best for me “stay in my current university as an RSP ” or “ get the post-doc position in the more reputed university”?
I think you need to think about what that post-doc position can really offer you. Going from a 25th-30th University to a 9th-12th University isn't that significant, since the precision of ranking schools is iffy, at best. Therefore, think about what opportunities the post-doc will offer you: exciting research? Connections/contacts? Does the PI have a solid track record of post-docs getting tenure-track positions? On the other hand, being a non-tenure track research professor may be the better option if you think you can get enough data to apply for a grant. The bump in income is nice, too, especially given the high cost of education in the USA. I'm guessing you probably don't have student loans from grad school, but what about undergrad?
tl;dr- Post-doc:may lead to tenure-track job, possibly better contacts? Research asst prof: may lead to tenure-track job, but with better interim salary.
- Scientifically you will advance more if you change the lab (also grant proposal reviewers, hiring panels etc will pick on this if you don't change lab after PhD)
- Position-wise it might be better if you stay with your former supervisor (if he is influential and might be able to get you a tenure-track position at the same department eventually)
Similar to above answers, what do you really want from a post-doc? It should be a learning experience to get to a more senior academic position. University rankings is not everything (though I know the US thinks strongly of this). I think it is also important to consider the lab, what they offer, or their reputation. Many top famous professors do not work in the absolute top universities.
Your current advisor is offering you quite a good opportunity with high independence and applications to funding, both important and good to show you can do if applying for a higher position in future. This is in addition to the benefits you mention of higher salary and good relationship.
I would only consider moving if you think the new lab can offer you ideas intellectually that may allow you to develop something you are interested in, in future.
PS - on a personal note, I declined a position at MIT for a global top 40 university, as I was interested in the professor, rather than the institution. So my suggestion is partially based on my own experience.