This seems to be a simple question but turns out to have a quite unexpected answer. The correct version is: elle s'est dit, and NOT elle s'est dite.

The explanation is, of course, that in French all things that seem to follow rules are just there to lull you in complacency. The amendment to the rule being that if there is a preceding COD in the sentence, you accord the participe. If there isn't, it is in pronomial form and hence invariable. An example:

Elle s’est dite intéressée par cette idée -- She said she was interested in the idea. In this sentence, the reflexive pronoun "se" is in fact a COD, hence you need to accord the participle.

Elle s'est dit que ça n'était pas la bonne idée. -- She told herself that this wasn't a good idea. Here "s'est dit" is a pronomial verb, which remains invariable.

You can read more about this topic here: https://www.rtl.fr/actu/debats-societe/elle-s-est-dit-ou-dite-7800087843

Hi Chris, just to let you know, I really appreciate your responses to people’s questions. Your answers are always so clear & knowledgeable, and really target what I believe is the crux of the question in each case. I gain a lot of insight from them, so - Thanks!

I found section five in this lesson very helpful:

https://www.leaflanguages.org/french-grammar-reflexive-verbs-passe-compose-past-tense/