Immigration records for the 17th Century are pretty spotty. The only "database" is Ancestry's "U.S. and Canada, Passenger and Immigration Lists Index" which should be used with caution. It contains a lot of secondary sources which are not necessarily reliable. I would say that if there were an immigration record related to John Harris, it would already have been noted in the NEHGR article noted in his profile.
NEHGS's Great Migration Project only covers immigrants with evidence of arrival by 1640, and it looks like your John Harris probably arrived later. NEHGS also has the Early New England Families Project which is working on developing sketches similar to the Great Migration, and eventually, they should release a sketch that might have more information. You can keep an eye out for new sketches here: Early New England Families FSP
I'm guessing that you're looking to find his English origins. Especially with a common name like John Harris, trying to find the evidence to connect John Harris in New England to any record for a John Harris in England is going to be difficult. There are a couple of ways that English origins are newly discovered. As more English wills and other records become available online, there are new discoveries of mentions of relatives being in New England. The other is looking for clusters of vital records that match up with a family. Unfortunately, the only lead here is possibly a daughter born in England (and Elizabeth is also a very common name).
Personally, if this were my brick wall, I would start with looking for early associates of John Harris, and those who arrived in Charlestown around the same time to see if any of them have proven origins. I would check those areas looking for baptisms of an Elizabeth with a father John, also checking for Harris wills. Unless you get lucky and find that "smoking gun," it means developing a list of candidates rather than an actual discovery. It's important to go into a search like this being aware that the answer may not be out there, and that the records just no longer exist...but there's no harm in trying.
Also, since this is your paternal line unless you have reason to suspect that you have a non-paternal relationship in your ancestry, I'd suggest taking a Y-DNA test (or see if anyone from John Harris's line has taken one). That still takes a little luck, but if you get a match it might give you the breakthrough you're looking for.
Good luck in your search!