Coming back to this thread a couple of years down the track, and much further up the track ancestor-wise, I think we've probably hit the wall with the Somerset Balnes, but thought it worth one more try with the earliest ones found so far.
These are a couple called James Balne (c.1680-April 1716 - burial at Newton Saint Loe) and Mary (c.1680 based on childbearing years). The only other sources for this couple are as parents of their children, from Mary (1701), James, Giles, Sarah, Esther to Anne (1713) on baptism records.
Daughter Mary (1701) is the first Balne recorded in Newton St Loe in FreeReg; earlier ones are all in Wellow (about 6 miles away). The earliest Balne in Somerset is in Wellow in 1562, and searching the adjoining counties yields nothing at all (up until 1700 - I didn't look later).
The other Balnes are ALL around Snaith in Yorkshire, and I cannot spot a single James up there - and actually none at all in the 16th C on FreeReg - the earliest are 17th C.

I think that I've now exhausted all of the usual sources and a couple extra via Google and local Somerset sources, so probably venturing into the realms of speculation here, but I'm interested in any suggestions or opinions. I am now wondering whether the Yorkshire Balnes (pronounce Bawn, like the town
https://en.wikipedia.org/wiki/Balne) arose independently of and later than the Somerset Balnes (pronounced
Ballin in my family line). Is it a coincidence that there were mineral baths at Askern (near Balne in Yorks) and Bath (near Wellow)? (Latin
balneum.)
And any evidence of James' birth (perhaps in Wellow?) or his wife's surname would be a bonus. I have truly given up on this one now though, so not expecting much!