Unless it is of a more Irish dedication, as I suspect is the case with Ms. Daly. Tyne is in fact her middle name, so it may well be a surname styled as a given-name. Granted, this is just personal conjecture. I always assumed her forename was as Irish as her last.
The Gaelic form would be "Teimhin" meaning 'dark,' 'dusk' and 'gloom,' it is typically anglicized "Tyne" or "Thynne." There are a two diminutive forms "Teimhnín" and "Teimhneán" which are both made "Tynan" or "Tivnan" in English. As far as I know, these English & Irish forms are now completely unused as given-names, alive only as surnames.
Further, the river named "Tyne" does not mean 'river,' but is likely from the same 'tun' root (meaning 'enclosure') as the word 'town.'