In Gail Carriger's series, mainly Finishing School, there is a female character named Sidheag, and I have no clue how this is pronounced. (Is it a real name in the first place or is it made-up?)
According to the author's website
Sidheag ~ SHEE-ak alternatively pronounced SIDH-hey, just to make matters confusing. (Again, chosen for its meaning, all the pack names are.)
The word itself means "Wolf" in Gaelic and is, apparently a traditional name in Scotland although it's never broken into the top 10,000 baby names in the last 100 years.
I know that Sidhe is pronounced "Shee" kinda like the e in eh, rather than the e in be. Technically DH in Gaelic makes a sound like what would happen if you drew out the GH in ugh, or if you voiced the CH in Scottish loch/German ach. However, a broad DH like that is sometimes just dropped at the beginning or end of a word, so you could probably ignore it so long as you get the rest right.
The e and the a in Sidheag sort of run together lightly, meaning that when you're saying it aloud your jaw drops a little and your lips go from the wider E configuration to a slightly rounder version in order to make a soft A sound. Not too much though– it's all rather subtle– and remember you're saying this all in one beat, so it should flow smoothly.
Finally, Gs in Gaelic are velar (with tongue to soft part at the back of the roof of your mouth) like the Ks in English (eg. kook)
I'm not Scottish, and I'm definitely not the best Gaelic speaker out there (I only know a few words) but I do my research. I hope I've been helpful.