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My swiss mother has lived in the UK without restriction since 1970. In 1970 she received a letter from the Home Office stating that restrictions on her stay here had been removed. In one of her old passports there is an ILR stamp (not in more recent ones).

She now wants to apply for dual citizenship and get a British passport. Our confusion is whether she first needs to apply for a Permanent Residence Card or not?

Form AN (page 29, Section 4 and 5) states:-
"SECTION 4 Evidence of settlement for applicants from Switzerland or the European Economic
Area or their non-EEA direct family members see pages 10 to 11 of the Booklet AN
• Your valid passport or valid EEA national identity card as evidence of your nationality; And
• A permanent residence card issued by UKVI

SECTION 5 Evidence of freedom from immigration time restrictions:required for all applicants
except those covered by SECTION 4 above.
• Your passport showing permission to remain permanently in the UK Or
• The Home Office letter by which you were given permission to remain permanently in the UK
• If you came to the UK as an asylum seeker you should have evidence of appeal applications
Or
• Evidence of being freely landed"

I am a little confused by this. Can she apply straight away using Form AN (she is language test and life in the uk test expempt), or does she need a permanent residence card first?

If her ILR has not lapsed, then I think she doesn't need a permanent residence card.

Commenting from my own experience, my "ILR" was issued in 1985 (HO letter and stamp in passport and residence card), and no gap in my residency. I applied just with the letter and endorsed residence card without any issues. Even phoned the HO Helpline where they confirmed this was sufficient.
So submitting your mother's original letter from the Home Office and endorsed passport should be fine.