Characters for ben1 ben1 meaning 围巾 or 围嘴?
It is just a guess:
围巾 is a scarf; 围嘴 is a bib, also known as 圍兜.
围巾 and 围嘴/圍兜 are two different things.
But the pronunciation /ben1/ gave us a clue. You are probably thinking of 'bib'
Bib  ( 圍兜 )
'Bib' sounded like 'ben1' in Mandarin. What you are looking for is probably a loaned word for 'bib'
If it is a loaned word, the character itself might have nothing to do with the actual item.
The only character that is pronounced as/ben1/  in Mandarin is 奔 /ben1/; /ben4/
It is just my guess, but to indicated  this loaned word  奔 is related to 'cloth' a 巾 radical might be added to 奔 and make it looked like this:

or this:

user3306356 wrote:

衯衯 could also be a possibility.

Yes, most loaned words are just sounded like the original foreign words. It could be some other similar sounding words in this case.
More importantly, I have no proof that it is actually a loaned word. We need someone who know the history of this item to give a definite answer.
Edit:

In Cantonese, 'bib' is called 口水肩--  often shortened to just '肩' /gin1/ , which is the only single character that can mean 'bib' in Chinese (Cantonese).
'肩肩' would be a baby talk.
However, it is pronounced /jian1/ in Mandarin