thumb rest ?

Its called a tug bar. I believe this came before the thumb rest.
I think the idea is you hook your fingers under it and then pluck with your thumb.

 

You are going to get a lot of wanky replies because this question is ask once or twice a month, but the simple answer is that it is a finger grip. You hold on to it with the fingers of your right hand while using your thumb like a pick.

Leo Fender put it on the original P-bass because he felt users of that early bass (Monk Montgomery and others) would find it useful. Later, in the mid 70's it started appearing above the E because many fingerstyle players moved it as a modification and CBS followed suit.

 

I got nothin.

 

I might add that AFAIK, current Fender models all put the finger grip back in the pre-CBS position below the G string. Others may be able to point out exceptions.

 

Oh right, it must be Friday 'cause someone's asking the thumbrest question for the thousandth time!

 

A clarification. Post CBS Jazz and Precisions had the finger rest (or as some refer to as the tug bar) on the lower side through the early 70s.

 

And you get an AMEN brother.

 

Asked and answered about 1,000 times here. Search and ye shall find.

 

You're probably right about most of them, Jim. But one exception is the AV '75 Reissue Jazz, which has it above the E string.