Does rel="home" on an anchor tag (<a>) do anything helpful?

The rel="page" was part of an initiative to create permalinks (see section 'Permalink detection') as part of a standard in HTML 4.

However with HTML 5 it now has no purpose and does not offer any accessibility or SEO value. It also might not validate using W3C validator anymore (not tested).

rel="something" should only be used on <link> elements, with the exception of rel="noopener", rel="nofollow" or rel="noreferrer" on anchors (<a> tags).

Note - There may be other rel="" for hyperlinks but the two stated are the only ones I can think of, it is no longer valid to use it for page locations, bookmarks etc.

Update

Thanks to @Sean who pointed out in the comments other elements can accept rel="", however MicroFormats are not the preferred way of adding structured data according to Google and their development is not as full fledged as using https://schema.org and JSON+LD.

“We currently prefer JSON-LD markup. I think most of the new structured data come out for JSON-LD first. So that’s what we prefer.” - John Mueller

I am obviously incorrect in what I said as it is perfectly valid, however personally I would not bother and stick with what Google prefers apart from the few items I listed.

See @Sean's answer for a bit more info on the subject.

for clarity rel="" has no bearing on accessibility