Often requested in sorrowful circumstances:

Ar dheis Dé go raibh a n-anam "May their souls be on God's right hand"

Notice that anam is singular. Why not anamacha (plural)? Because each person has only one soul.

Also:

Ar dheis Dé go raibh a anam "May his soul be on God's right hand"
Ar dheis Dé go raibh a hanam "May her soul be on God's right hand"

(Corrections welcome.)

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Múinteoir Gaeilge - Irish Teacher
My "specialty" is Connemara Irish, particularly Cois Fhairrge dialect, but I can also speak Ulster and Munster Irish with native-level pronunciation.
Is fearr Gaeilge ḃriste ná Béarla cliste, cinnte, aċ i ḃfad níos fearr aríst í Gaeilge ḃinn ḃeo na nGaeltaċtaí.
Gaeilge Chonnacht (GC), go háraid Gaeilge Chois Fhairrge (GCF), Gaeilic Uladh (GU), Gaelainn na Mumhan (GM), agus Gaeilge an Chaighdeáin Oifigiúil (CO).

There's nothing to correct :-)

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Is fearr Gaeilg na Gaeltaċta ná Gaeilg ar biṫ eile
Agus is í Gaeilg Ġaoṫ Doḃair is binne

Is this something that would be used when referring to a person who's passed? For example, in my dad's family, the older members will often say "may God rest his/her soul" after speaking the name of someone who's died. Would this be used similarly? Thanks!

No it's normally used more formally.

Go ndéana Dia trócaire air/uirthi
or
Go dtuga Dia suaimhneas dó/di

would be a more informal way.

The first is more formal and is used more often in death notices and on memorial cards. What Chucky gives is used when apeaking of the deceased