The problem with digital cameras and flashes is the "pre-flash". The camera sends a low power flash just fractions of a second before the shutter opens so it can calculate the proper exposure.
Film cameras don't have this issue because the film is reflective and the metering system can read the reflection off of the film surface in "real time".
Digital cameras have not been able to duplicate this process yet.
It's the timing between the pre-flash and the shutter opening that causes some people to blink. If the time between the pre-flash and flash is short enough the person doesn't have time to blink.
It all comes down to the persons reaction time.
With my last camera I would get my daughter blinking about 75% of the time. My son was never blinking.
With my new body I never get any blinkers because it has a much shorter pre-flash delay and people don't have time to blink.
Some digitals have the option of forcing a pre-flash - you pop the preflash, the camera does the calculations, and when you're ready you take the picture.
You can also avoid the pre-flash issues by using the self timer on some cameras.
I used to set mine to a 2 second delay. The pre-flash would happen when I pressed the shutter release, and then seconds later the shutter would open. Worked fine for portraits and things but wasn't very practical.
Here are some sample pre-flash times as examples
Times are in seconds.
.190 - KM A2 w/5600HS ext flash
.130 - .150 Konica Minolta 5D/7D, Sony A100
.130 - KM A2 w/internal flash
.119 - Nikon D50
.116 - Nikon Coolpix 4100
.115 - Nikon D70
.106 - Olympus C7070
.100 - Epson 3000Z
.084 - Sony Cybershot dsc-p93a
.080 - Nikon D80 (lag timing)
.068 - .070 Rebel 300D
.063 - .065 Canon SD200
.052 - Canon SD500
.052 - Canon S2 IS
.050 - Nikon D200 (lag timing)
.037 - Nikon D2HS/D2XS (lag timing)
As far as blink speeds;
.100 seconds - Time from preflash fire until there is definitely a slight change in the eyelid indicating the beginning of the blink, yet not quite a lazy eye look.
.117 seconds - Eyelid starting to droop more noticably / Lazy eyes
.133 seconds - Eyes mostly closed
.150 seconds - Eyes completely closed
It also takes a while for the eye to reopen, so an even slower preflash will still catch closed or lazy eyes.
I'm afraid I don't have any specific recommendations for you, but if you're looking to replace your camera see if you can find a pre-flash measurement in the specs or on a camera review site.