I've done a little poking around on the Belgian side, but without much luck.
The Royal Archives in Belgium have only recently started to digitise their records.
Here's the link:
http://arch.arch.be/component/option,com_analyses/Itemid,122/lang,fr_BE/(The button promising English actually delivers Flemish, so I'd stick with the French...)
No Boniverts of any kind.
Essentially, there is no national registry online, and as far as I can work out, the fiches at the Royal archives are commune based, not surname indexed! In other words, you need to know what commune your relative was born in, in order to find the Birth Certificate.
If you want to keep looking in Lille, just to be sure, the document you referenced with over 1000 "naissances" (yes, births) is in alphabetical order. You'll also need to look in the surrounding suburbs, as "Lille" probably just refers to the centre of the city. (google maps would probably give you some ideas )
The information from the website genealogy.tm.fr essentially says they have only a handful of records in 4 departments (=counties, more or less). They're including Belgium as a county! In Belgium, there are only 3 births, between 1679-1750, so nothing worth spending your money on.
However, here's what I would suggest:
This geneaology website
http://www.genealogie.com/v4/forums/recherches-genealogiques-mains-tendues-f2.htmlis the closest to Rootschat in friendliness in French I could find, and they have a section called "Mains tendues" (helping hands) (which is what I've given you the link for). Join up, and post your query. Google have a good translation tool (translate.google.com), but if you get stuck, but if you get really stuck, PM me and I'll translate.
Here's your query, in French.
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Je cherche à trouver l'acte de naissance de mon arrière-arrière grand-père Joseph P. Bonivert (ou parfois: P.J. Bonivert, Pascal Bonivert, Paschel Bonivert, Peter Bonivert).
Il est arrivé à Chicago aux Etats-Unis le 13 aout 1864 par le bateau Favorita, partant d'Antwerpen. Il donnait comme adresse "Lille" mais plusieurs de ses documents pendant sa vie aux E-U montrent son lieu de naissance comme la Belgique.
Comment ferais-je les premiers pas pour trouver son acte de naissance ou d'autres signes de son état civil? Je vous remercie tous de tout aide que vous pourriez me donner.
(Comme je ne parle pas le français, ce message a été rédigé par une amie, alors pardonnez-moi si je vous réponds lentement, car il va me falloir lui demander une traduction... Si jamais vous pourriez répondre en anglais, cela me ferait énormément plaisir.)
Je vous remercie tous d'avance de tout aide que vous pourriez me donner.
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If you plug that into translate.google.com, you'll get a pretty good idea of what it says :-)
Good Luck!
