1 uSv of Am-241 with no shielding produces an exposure of ~1.27 Sv per year assuming constant exposure at a distance of 1m. That is less than 1/700th of the recommended annual exposure limit of 1000 uSv annually. So "a little bit" in this case is almost zero which is why it is used in smoke detectors to begin with.
Ah, I hadn't realized such a small quantity will suffice to get your RNG working.
Well it remains to be seen but that is my hypothesis. The reason for this specific tube (LND 712) is that it is very sensitive to alpha emissions. The source is going to be permanently attached to a screen on the window end of the tube. There are more sensitive tubes but they are out of what I am willing to spend on a hobby project. Looking at the test results of other homemade geiger counters it looks like 1 uCi of Am-241 will register 100K to 120K CPM with this tube. Assume two counts per bit that works out to ~800 to 1000 bits per second peak throughput.
The source or the tube however isn't going to be the bottleneck (at least through 1 kbps). The hard part is going to be getting a timing circuit which can register events with sufficient accuracy. We are talking an average interval of 500 microseconds so a timer with microseconds scale accuracy (or at least tens of microseconds) is going to be necessary. This is beyond the capability of most micro controllers, and it probably going to mean a dedicated real time clock ( something like
http://www.maximintegrated.com/datasheet/index.mvp/id/4627/ln/en ).
As a proof of concept I am going to start out without a RTC but that means much lower timer accuracy and lower throughput first. Something in the order of <3,000 cpm which produce ~24 bps of entropy. Even that will depend on micro controller having true 1ms clock accuracy. For the early test I am going to use a gas lantern mantle (thorium & beta emitter) as the particle source.