Don't know Somerset terribly well but if at all possible, avoid having to use the M5 at junction 21, Weston Super Mare at all costs during rush hour morning and night...held up round there sooo many times. Good luck with your hunt.
What do you want it for? do you need good hacking? Or access to events? Avoid the levels if you want dry fields and/or hills to hack up.
Essentially Exmoor or the quantocks IMO unless you need better transport links, though there are nice bits elsewhere too and will depend on budget.
chillipup junction 21 used to be miles away from actual weston, and they have essentially built commuter houses all the way out to the motorway and surprisingly the junction can't cope. TBH anywhere near the M5 is an issue all summer! and the following two junctions (22 and 23 have also had a lot more houses built near them of late though I think they cope better with it mostly atm).
The Blackdown Hills (Devon/Somerset border area) or Quantocks. Blackdowns are south of the M5 and Quantocks north - both handy for junctions onto M5 at Wellington or Taunton if that's a factor. There are some lovely bits of south Somerset (further east of county) but nearer you get up towards Glastonbury/Wells/Chew Valley area it gets expensive for cost of housing etc, and busier roads.
Agree with Quantocks, Exmoor and Blackdowns area. I live in the worst part of Somerset for hacking, unfortunately, there is no off road riding whatsoever! I was stunned, given its such a rural county! So double check the OS maps and ask locally before you buy.