Sure, Avril was rarely used in the US until Lavigne came along, but in Britain, it seems to have been a slightly common name, which started seeing rising usage in the 1910s & 20s and peaking in the 1940s (there are more than 4,500 records on the England & Wales Birth Index for that decade). It wasn't until the 1960s that April took over as the dominant form and, nowadays, April is far more often used there than Avril.
April might be an English name, but Avril certainly isn't a French one. No one's called like this in France, and by the way it would seem male, like the month. Avril Lavigne might have a French sounding name but it's nothing French at all, I'm afraid. You're free to call your child Avril if you want, but in France, it would be very unusual, and your kid would get teased about being called like a month of the calendar. ;)
I'm French, I live in France and I can confirm that Avril IS French. I know of several Avrils and they never get teased. There are about 638 Avrils in France and the name has been given since 1950.
You are right it's NOT a French name, it's ENGLISH - deriving from the Anglo Saxon Eoforhild if you cared to read the post above yours. It quite explicitly states that. Avril is only used in English speaking countries although with the singer in mind and having done some research it does get some use by French Canadians and obviously was used on some males in France (and South Africa).
― Anonymous User 10/29/2008
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Well, the entry for the name still states Avril as a French name, which it isn't, so please spare me...
Sources: https://en.m.wikipedia.org/wiki/Avril_(name)
https://forebears.io/x/forenames/avril