In lieu of any definite information resulting from your post here, I'd suggest purchasing Edith Anne's birth certificate to get her mother's maiden name.
The parents of Edith Thomas baptised in Begelly in 1876 per Orielbenfro's post are not the same as you are looking for, these were George and Margaret.
According to FreeBMD there are two Edith Annie Thomas births registered in the Pembroke district which covers Tenby. Based on Orielbenfro's post, the one in 1874 appears the one to purchase.
If I were to speculate on her maiden name, I'd follow the Davies idea too. There's a Mary Davies living as a visitor with Thomas/Edith Annie in the 1861 census of the right age to be her mother. In the Begelly baptism register there's an Ann Davies baptised 6/10/1833 to James & Mary of Wooden, labourer. Wooden is an area split between St Issells and Begelly (East Williamston) parishes so a possibility. But it's a long shot.
Edith Annie's birth certificate seems the way forward unless SKS can find her in the Tenby parish or non-conformist registers.