How could this small lost colony survive? and would it change the history of the United States or it would lose relevance during the following centuries?
It was officially the Citte (City) of Raleigh in Virginia, which might clue in on its history-to-be.
It's essentially Virginia, just twenty years earlier and under a proprietor (Walter Raleigh). The loss of Roanoke in OTL hurt his finances, while here it's more spread out but possibly more damaging in the long run for him to keep resupplying and resupplying it. Indeed, Roanoke's failure damaging his money matters so much helped develop the concept of the joint-stock company.
He got beheaded in 1618, and in OTL the Virginia Company sold it to the Crown in 1624 due to loss of profits, so I have a distinct feeling the same history as in OTL is played out in slightly different circumstances - the Crown takes the colony over from our headless hero and the joint-stock company idea develops anyway due to seeing what a drag it did to ole' Walt.
How would it survive? Well, plans were to re-locate it about 100 miles north, which would place it more or less where OTL Norfolk is (NOT Jamestown or Williamsburg - this is important), and since Norfolk as a metropolis exists in OTL, 'Roanoke/Raleigh' City would probably live on as a major city. For economics, Raleigh helped popularize tobacco - so assuming the same people who settled Jamestown merely settle on the other side of Hampton Roads while history otherwise plays the same for them, tobacco will finally give Walter some breathing room (and save the colony). Jefferson will still relocate the capital to Richmond in 1780 due to 'Roanoke/Raleigh''s coastal location, but if Norfolk survived the OTL 1775 Burning, it'll bounce back from the capital move as well because it's such an excellent port and shipyard (and it just got BURNED, it really can't get worse than that). Which means the literal First City of America is still a major burg, which means the southern USA has a city with as strong a sense of history and culture as Boston has for New England and the North-in-general, which could have a fun effect of historical perspectives for the country.
Would it be a bigger city than Norfolk on OTL is? I would assume that it could probably reach the sizes of some of the northern cities, if immigrants to Virginia decide to set up shop there.