I was named by a monk from a shrine that my family often visits.
A monk from a shrine that my family always goes to gave me my name.
この場合は、次のように言うと良いですよ。
ーI was named by a monk from a shrine that my family often visits.
「私は、私の家族がよく行く神社の神主に、名前をつけてもらいました」
または次のような言い方もできます。
ーA monk from a shrine that my family always goes to gave me my name.
「私の家族がいつも行く神社の神主が私の名前をつけてくれました」
ご参考まで！
I was given my name by a monk at the shrine that my family frequents.
ご質問ありがとうございます。
Shikaさんの英文を少し直します。下記通りです。
I was named by a monk at the shrine that my family often visits.
上記の英文で十分ですが、別の自然な言い方を知りたいなら、「I was given my name by a monk at the shrine that my family frequents. 」でも可能です。
ご参考になれば幸いです。
カナダ 
アメリカ合衆国 
