The Roman Map of Britain Eiuden(s) Ayton, Berwick, Borders or Inveresk?
Euidensca var. Eiudensca (R&C 189) next
Eiuden(s) is the
proper form, -ca belongs to the next entry Rumabo for Carumabo
(R&C 190), the final -s is
likely a text divider or an -is abbreviation, i.e. Eiudenis.
The EyeWater is recorded as Ei 1098 and Ayton Eitun 1098. In the Cosmography's river section EyeWater is perhaps recorded as Viuidin (R&C 261). Ayton is only a place-holder for want of a better answer.
But there is what appears to be another Eiuden(s) found in Bede 'Urbs Giudi', later 'Iudeu' in Nennius' Historia Brittonium, and in Y Gododdin the Firth of Forth is described as merin iodeo 'sea of Iodeo'. Jackson* examined Urbs Giudi as a choice between Stirling or Cramond, though neither can be it. Stirling is north of the Antonine Wall, and Cramond is Carum, the next entry. So we could have an earlier name of Leith Inverlet ca.1130 (NT2776) or Inveresk. But Leith(en) appears to be an old Celtic river-name compared with W. llaith 'damp, moist', lleitho, 'moisten, overflow'; so we are left with Inveresk (NT3472). The -esk of Inveresk is not necessarily British isca, but perhaps Gaelic uisge.
*Jackson, Kenneth H. 1981. "Varia: 1, Bede's Urbs Giudi: Stirling or Cramond?." Cambridge Medieval Celtic Studies 2:1-8.