Because of its construction, pointed nib pen, Bodoni's sans counterpart is something in circle of Helvetica and other "swiss" typefaces.
Massimo Vignelli had some good combinations of Bodoni and Helvetica - especially in period of 60's and modernism.

Avenir and Avantgarde, that is a bit tough, because they are geometric fonts, and first answer would be Bodoni or anything in didone area. Avantgarde has its serif compatriot in Lubalin Graph, slab serif version of avantgarde.

I think it's also important which Bodoni do you use. If i think of Bauer Bodoni No.2 I'd better try Akzidenz Grotesk instead of Helvetica. I think a lot of older static Sans Serif typefaces can work in one way or another, Franklin Gothic for example..

Akzidenz Grotesk is great choice, and yes, it is important which Bodoni. Also you could risk with Baskerville as serif choice because of its transitional character.

Bringhurst suggests several traditional combinations. Off the top of my head:

Gill/Perpetua
Futura/Bodoni
Frutiger/Meridienne
Helvetica/Craw Clarendon
Syntax/Minion

I have had success with Helvetica Neue and ITC Century, especially due to the fact that they are both large families. The trick when combining serif's and sans' is to counterbalance the x-height, stroke width, character width, ascenders, descenders, cap height, axis &c. That's a lot of different variables. Look at this thread on Utopia and Univers for a good Illustration. The traditional school of thought says go for similarity or contrast, but not in between (shit or get off the pot). Utopia/Univers is a good example of similarity while Futura/Bodoni is a good example of contrast. If you can afford to, I highly recommend investigating Berthold's Futura Serie as it has small caps and old-style figures.

Some typefaces, such as Rotis, have both which can be convenient. And recently, I noticed that Font Bureau has a similar/compatible list of typefaces on the right hand side of their specimens. Also very convenient.

At the end of the day however, it's not the font, it's how you use it that counts.

(but that's just something people with small fonts say)

I went through a phase where I used Futura with Century Oldstyle a lot. They don’t have a lot in common, other than a sturdy, no-nonsense appearance, but I do like the way they look together.

Bless you, my beamish Victor!