How long should I register a domain name that will be referenced in scientific publications? I have a website (with data and modeling interface) that will be referenced in a series of scientific publications that are beginning to be published.

Currently the website is along the lines of myserver.institution.edu/mysite but I would like to change this to something like mysite.org.

Costs for the site range from $37/year for 2 years to $9.99 for 100 years. Even 100 years would be a tiny fraction of the overall research budget, but my first inclination is to go for five years at $33/year. But I don't even know if the web will still be around by then, much less a server hosting this work.

(this is related, but not a duplicate of related, but not a duplicate of How to archive an academic blog or website?)

Is five years long enough? Is 100 years justifiable?
If you consider the data of permanent value and essential to your paper, then you should try to preserve it permanently.  The question in that case is not how long to register a domain name for, but rather how to find an organization that's willing to archive the data in the long run.  If you don't have that, then it's going to disappear eventually anyway.  If you do, then you can do whatever you want with the domain name as long as you provide enough information that someone could access the data from the archive even once your domain has expired.

Most scientific publishers solve this problem by archiving supplementary information and data along with the paper.  There are also field-specific data banks.

On the other hand, if this isn't an essential part of your paper (but rather just for illustrative purposes), then you can do whatever you want.