I am trying to find a word which describes someone who is both old and wise. I would like the word to describe someone who is extremely frail and infirm. It might also have the connotation that their counsel is odd or hard to follow, even if it is sound advice.
The closest word I can think of is "sage", which seems to connote someone who is both old and wise.
In certain contexts, it might also be possible to use "patriarch" or "elder", which both imply wisdom or age, but these words add some political connotations that may or may not be desirable.
I'm not sure if there exists a word that includes quirkiness, too. If you don't mind using semi-proper names (in a somewhat more metaphorical sense), you could use "einstein", "oz", or "yoda". These can be probably be used as common nouns, but they may be too strongly tied with the original individuals the words come from.
I am trying to find a word which describes someone who is both old and wise.
The word you are looking for is venerable, which per the OED is:
Commanding veneration or respect by reason of age combined with high personal character and dignity of appearance; having an impressive appearance in virtue of years and personal qualities.
Even if that is not quite exactly what you are looking for, then you should be able to find related terms easily enough.
Regarding the other aspect you asked about, that the advice be odd or difficult to follow, you might start with Delphic. The OED gives one sense for this as "oracular, of the obscure and ambiguous nature of the responses of the Delphic oracle", and provides as a citation the following:
This reads rather like a Delphic response.
The term doyen is often used to convey some of those characteristics. According to Merriam Webster:
a : the senior member of a body or group b : a person considered to be knowledgeable or uniquely skilled as a result of long experience in some field of endeavor
Often this term is used with "the" to indicate the most knowledgable or skilled person in a field. According to the Compact Oxford English Dictionary:
the most respected or prominent person in a particular field
This term probably does not convey frail, infirm or quirky.
SUPPLEMENT: Another possibility is a phrase, not a single word. The term ancient one connotes both age and respect, maybe wisdom, especially when used as an address, and has a feeling of frailty. I still don't think it conveys quirky, though.
Venerable? (No quirkiness) Wizened? (No quirkiness) Sensei? (In its English usage, anyway.)
My guess is that you could swing it with two words, but you wont find one that completely covers it.
Since you're looking for a single word, I assume you're writing something and not trying to explain it in conversation. If so, I'd say make up a word that sounds right, and tell the reader that it means or has as connotations all the above things.