Introducing Modern  

Classic Bergeres chair...you can have a blast with these!
here's some thoughts...
Paint the wood high gloss white lacquer and upholster in faux zebra
Paint the wood high gloss black and upholster in a good quality velvet in a color that works with your space
Paint the wood metallic silver and upholster in white leather
Paint the wood a rich lipstick red and upholster in black leather
Upholster in a classic Maharam or Eames print and strip the wood to a natural oil-rubbed finish if it's in good enough shape or paint it a 'natural' brown/tan color
The trick is to use a finish that is totally unexpected forthe shape of the piece and to keep the shape of the chair noticable by using a high contrast of the exposed wood and the upholstery.

Two and half cents
Its a classic piece, I say keep it simple either with a pop of color or a netural fabric. Nothing like you're trying to hard to "blend." I found this one, doesn't it look fab?!

I actually have wanted to do...
I actually have wanted to do this for quite a while. Just no space for another chair yet, and none I'm willing to part with. Have a ball. I love the gloss white or black / unexpected upholstery option.

So you've got some elegant...
So you've got some elegant pieces (bellini and saarinen) and some modern pieces (saarinen and Eames) and some playfulness with materials (Starck). I would continue the playfulness with materials in any of the ways Olive suggested. The Zebra/Gloss white, or gloss white with grey Eames Dot fabric are my favorites. Pair it with the Saarinen table and you've got a great arrangement.
I'd keep the chair on the left; its got way more personality than the one on the right. Whichever fabric you use I'd get enough to make a throw pillow or two to bring it back again somewhere else in the room. Whatever you do, use a classic fabric, and not some crazy faddish thing (thinking bright green leaves here) that will need to be redone in 12 months.

Great Ideas
Please keep them coming! Love the orange bergere and the one on the right is leaving for greener pastures this weekend. I like the idea of painting the frame.
Also, I was going to post this in another thread but I might as well add it here. I have a victorian chaise with great lines AND I HATE THE FABRIC! I don't have the money to upholster right now (as I would rather do the bergere)and I managed to find a wall color that works well with it. How can I hide this funky fabric for now? Pillows? Throws? Thanks for the help.

Embrace the paisley
It's a classic pattern and a popular one for the period of the chaise...add fake fur pillows or something like that to add a modern contrast and just love it as is until you can reupholster. That piece will also look good done with a modernist's eye.

Ditto with the paisley, it...
Ditto with the paisley, it isn't my style but you can make it work as-is for now. Oh, and loving the wall color; I just painted my living room/ dining area that same tone.

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You have a good eye for shape,the one on the left is older 20s-40s? and a better quality chair.Chaise good form...The other chair is not that bad if you dont compare it to the left one...

I'm with you AzC
The one that she decided to keep has a much more graceful form...I can see it done up mod in my mond's eye and it'll be fab!

I read french interior deco...
I read french interior deco mags and often see a mix of antique and design classics. I've seen this type of chair done in silver, i.e. the wood is painted (or sprayed) a metallic silver with the upholstery in a similar fabric (grey silk for example or flannel); also seen wood white or black painted with a graphic black and white print...
I saw a 'banquette' of the same era done in white paint and white linen upholstery...

Fantastic
I love these ideas. I never thought about faux fur pillows on the chaise. I can't imagine anything else working with it. I plan on putting the black tulip table next to it.
Thank you for the compliment on the lines & form. I love curves in my furniture. And I'm a minimalist to some degree. I dislike cluttered surfaces or drawers. I get peeved when I can't find something so I start throwing things out. lol My family usually receives my cast offs and they love it.
The color on the wall is "Ralph Lauren - Strudel".
I haven't decided what to do with the chair just yet. I love red and I keep thinking red velvet but I would like to be more adventurous then that.

You don't need to get too adventuresome
The chair already had a great form so go ahead with the red velvet it'll be sumptuous. Black, dark chocolate, pewter or even citron green would all be interesting colors for the paint on the wood. Do what makes YOU happy not what you think is going to be perceived as 'adventurous!
And I do love room with a mix of old and new...more sophisticated...especially when the occupant can really curtare the space and make the sapes work without going over the top! Sounds like your the girl for that!

DLD: Look!
I found this website, check it out!
http://www.metrosofa.com/chair-inventory.php

teehee
marimekko upholstered seat cushion
white flokati pillow for the back