Should I host my academic website under my institution domain or under a domain of my own ? My institution provides a dedicated storage space to host our academic websites, which is reachable via the institution domain (something like http://people.institution.org/John.Doe/).

I try to have a tidy website of a few pages listing my publications, research area, contact, etc. It is currently reachable through my institution domain. I try to also put some efforts in SEO when people look up for my name or particular research area on search engines.

I'm in my last year of Ph.D. and I will undoubtedly move to other institutions in the next few years, so I am wondering if it wouldn't be preferable to host my academic webpage on a dedicated domain http://www.johndoe.com, which would prevent the need to do the SEO all over again each time I will switch to a new institution.

The only advantage to use my institution domain is the affiliation, but this information is present on my website anyway.

I'm a Ph.D. student in electrical engineering in France, if it matters.
Why don't you just do both? By both, I mean use both URLs. I "do both", so when I graduate, I'll still have my site for others to see. You can do this in many ways, but I had my university student page auto-redirect to my personal home page. The code for that is like a one-liner.

This grants me the opportunity to refer people to different sites depending on the situation. I think myname.com is undoubtedly easier to remember than the nuances in my university student site URL: people.school.org/first.last ... On the other hand, if the situation is more institution-based, perhaps it's better to stick with my college's name. You've got options this way.