The (kinda) new German Personalausweis has two - separate - uses vaguely related to "electronic signatures":
The "Online-Ausweisfunktion" (on-line identification), also called eID (for "electronic Identity" [sic!])
When applying for one, you will indeed be asked, if you want to use this feature. If you answer "yes" (and are old enough), you should get a "PIN-Brief" (PIN letter), containing
- a five digit PIN for final activation of the feature (which will have to be changed to a six digit one during activation)
- a PUK
- a password for deactivating the eID in case you lose the card!
Using this feature, anyone registered with and authorized by the German authorities can electronically "ask" the Personalausweis for certain information (like name and age of the bearer - the need for each piece of information will have to be proven to the authorities) and get a signed response from it, thus knowing that the information is correct.
The "Unterschriftsfunktion" (digital signature)
To use this, one has to acquire a singing certificate (subject to German Signaturgesetz) from one of the authorized suppliers of those.
Using this certificate, one can electronically sign things, which then have basically the same legal standing as physical documents under German law.
The services of Deutsche Rentenversicherung seem to need the eID functionality, not the signature.
So, what you need to do is
- Locate your "PIN-Brief"
- Use the "Transport-PIN" therein to activate your eID and set a PIN
- Use the eID with your PIN to use the services you want.
Everything depends on you still having this letter, though.
Update for the clarified intent of the question:
After researching a little it seems clear that, at least technically, there is a possibility to activate the eID function without the PIN-Brief.
As, according to this page for the Generalkonsulat Miami, which I suppose is the correct one in this case, a Personalausweis can be ordered there and the PIN can be set there.
As they have the technical means to do this, you should be able to set your PIN, when you - personally - take your Personalausweis to their location. You will probably have to make a date for that and pay them dearly.
The only two possibilities for changing a PIN seem to be "with a valid PIN or Transport-PIN" or "on-site of a German Personalausweisbehörde"!
Even though I'm quite confident that this information is correct, I'd still call them and ask, before driving there...