I should have clarified that he bases some of his advice on well-validated research. He also speculates well beyond what research will show, and doesn't hesitate to offer advice there, too. (Especially recently.)
Nonetheless, the core of his advice is (arguably) all based around well-validated correlations, even if causality isn't always established (and causality is what is important if you're giving advice).
I don't own any of his books, so I can't mine references from them. I don't remember the things he's cited or referred to, and it takes forever to find them because he's not very good at giving references when speaking.
So I'm going to kind of clobber together a mess of things because this isn't my area of expertise, and I can't reconstruct the validation steps I took before. Nonetheless, it's broadly consistent.
1. "Stand up straight with your shoulders back." (Which apparently means something like, 'be resilient' and 'accept challenge'.) Resilience is highly correlated with satisfaction: https://www.tandfonline.com/doi/full/10.3109/13685530903296698
2. "Treat yourself like someone you are responsible for helping." This generally falls under self-care, which is widely regarded as important, and there are a variety of strategies that improve well-being (e.g. https://link.springer.com/article/10.1007/s11205-004-0995-3).
3. "Make friends with people who want the best for you." Aside from avoiding people with dark triad traits, I couldn't find something quickly that supports this, but having close friends is a positive and that's presumably easier when they want the best for you (not very good support but, for example, https://link.springer.com/article/10.1007/s11205-011-9861-2/tables/3)
4. "Compare yourself to who you were yesterday, not to who someone else is today." This is basically a statement of a growth mindset. Lots of support for that one, e.g. https://www.proquest.com/openview/2987b657119a712ac019e668ba1e29d6/1?pq-origsite=gscholar&cbl=2029209.
6. "Set your house in perfect order before you criticize the world." (Can't find any particular support for this, but it is related to a call for resiliency and self-reliance, both of which are positively correlated with well-being.)
7. "Pursue what is meaningful (not what is expedient)." This has been shown in numerous studies; here's an example (not the one I had in mind, which asks elderly people, but I'm trying to get through this quickly to illustrate that there's support, not come up with a definitive resource: https://www.researchgate.net/publication/235331836_The_Relative_Impact_of_Personality_Traits_Meaningful_Occupation_and_Occupational_Value_on_Meaning_in_Life_and_Life_Satisfaction)
Specifically for young men, Peterson seems to emphasize things like, "Responsibility is what sustains you in your life." I can't quickly find what I've seen on this in the past, but here's a recent paper supporting this outlook for Chinese youth: https://link.springer.com/article/10.1007/s10902-019-00134-6
There's a "do something useful" advice too, which generally maps onto being gainfully employed with something important, and that is highly correlated with life satisfaction especially in men (above references cover this, but I can add more if needed).
If you ask: what factors correlate with reports of life satisfaction in men, you find (e.g. in https://www.tandfonline.com/doi/full/10.3109/13685530903296698 again) things including: resilience, employment, partnership, self-esteem, and income, most of which (all, if "set your house in order" includes employment) are core virtues touted by Peterson.
Peterson's research specialty is in Big Five personality traits, and a lot of his advice is just reading off the traits and saying to lean into the side that generally correlates with better outcomes.
Does this mean his views on social trends are always on target? Absolutely not. But is he generally on-target for giving advice on how to lead a satisfying life? Sure seems that way.
Anyway, this is kind of haphazard and random, because again, this isn't my area of expertise and I don't have time right now to chase citations to get to the seminal papers in each area. But I think it should suffice to show that there is substantial literature support behind a majority of Peterson's advice.