My current recommendation for a Flip-style camera is the new iPod Touch:
http://www.apple.com/ipodtouch/
It has a 720p camera and you can edit directly on the camera with a variety of inexpensive apps such as iMovie ($5). Fits in your pocket so it's always with you. Joby makes a great little flexible tripod case for it too: (wraps around the crib easily & goes down to the baby's eye level on a flat surface)
http://joby.com/gorillamobile/touch/
The larger question to ask when choosing a camera is workflow. How are you planning on storing the footage, editing the footage, and exporting the footage? Are you going to make DVD's to send to Grandma, or short movies to upload to Facebook? This is where a dedicated camera starts to get tricky, and why I recommend the new iPod for a quickie camera - low requirements for a support environment to actually spit out video projects. I have an iPhone 3GS (similiar video functions to the new iPod) and it's really handy with my 3-month-old because I always have it with me and don't need a computer to cut clips or do editing (although you can import the footage into whatever editing program you like).
If you
are looking for more of a dedicated camera, the Kodak Zx5 has pretty decent low-light:
http://store.kodak.com/store/ekconsus/en_US/pd/PLAYSPORT_Video_Camera__Zx5/productID.221644700
I have the Zx3 (similar older model), which I have since donated to my sister (due to the iPhone) and she takes it to concerts all the time with great low-light results. Amazon has them in a variety of colors for around $110: (they take SD cards; 32-gig cards are getting pretty cheap, such as a Transcend model)
http://www.amazon.com/Kodak-PlaySpor.../dp/B004FLL5BI