Her Japanese Wiki page tells us the origin of that nickname:
愛称の一つである「へごちん」は、2013年にweb番組『五十嵐裕美のチャンネルはオープンソースでっ！』にゲスト出演した際、「頭へ ゴチン」を「頭を ヘゴチン」と読み間違えたことが由来。さらに省略されて「へご」とも呼ばれるようになった。
My translation and my explanation in the parentheses:
One of her nicknames "Hego-chin" came from an incident that took place in 2013 when Ohashi made a guest appearance in Igarashi Hiromi (another Japanese voice actress)'s web series Igarashi Hiromi's Channel is Open Source! In the show Ohashi misspoke and mistakenly said "hegochin the head" (which is nonsensical) where she should've said "gochin to the head". She has also been called "Hego", a shorter version of "Hego-chin".
「ゴチン」(gochin) is an onomatopoeic word that describes the sound of a large object hitting someone's head. 「頭へゴチン」means to use something to hit the head in the fashion of "gochin". Here 「へ」is a particle indicating direction. The kana is usually pronounced "he" in words, but as a particle it should be pronounced "e". So the correct reading of that phrase was "atama e gochin", but Ohashi thought 「へゴチン」 was all together a word, so she pronounced it as "hegochin". Another tricky thing that led to her mistake was the hiragana 「へ」 and its katakana 「ヘ」 look alike.