Reading the question and the comments, there may be a conceptual misunderstanding : the attenuator WILL attenuate any noise presented on its input (even from just a 50 ohm source impedance), to the same extent it attenuates the signal.
However it also generates noise of its own, which may be represented as the noise from a perfect resistor equal to its own output impedance, and this is added at the output to the (attenuated) input signal and noise. So if input and output Z are both 50 ohms, the net result is attenuated signal + marginally increased noise (i.e. NF = attenuation).
But if its output impedance is lower, the added noise is also lower, thus improving the noise voltage as Andy states.
So represent the attenuator as a perfect attenuator (attenuating noise) in series with a Johnson noise voltage source equal to the output impedance. The rest is just applying the formulae.
EDIT: re: updated question.
(1) There is nothing special about 290K except that it's a realistic temperature for the operation of a passive circuit. The reason they chose it is that the article quotes a noise floor ( -174dBm/Hz) which is correct for a specific temperature : yes, 290k.
(2) While any resistance in the attenuator will contribute noise, I realise that it is not a satisfactory explanation as to why you get the same noise out of an attenuator, because (as Andy says) you could make a capacitive attenuator which is not a Johnson noise generator. So we have to look a little deeper, and remember these noise sources are the statistics of the individual electrons that make up the current.
So, let's say we build a (50 ohm in, 50 ohm out) attenuator, and attempt to cheat Johnson by using a capacitive divider. That implies a node within the attenuator which conducts some of the input current to ground. At this node, we have two current paths; a fraction of the current flows to output, the rest to ground. What determines which path an individual electron will take? Essentially, chance. Collectively? Statistics. So this is a noise source.
Or let's just add series capacitance to provide enough attenuation : we thereby avoid dividing the current flow and eliminate the noise source, right? At the cost of reducing the signal current; our statistics now operate with a smaller sample size and consequently greater variance : more noise.
These results are the best you can do, there is no way round them.
Your amplifier has average noise characteristics, the problem is that your signal is very, very weak. The amplifier is responsible only of the noise part of the SNR, so it has not a "poor SNR", but a "poor input referred noise".
To obtain a better SNR you can either amplify your signal, without adding noise, or reduce the noise.
Since amplifying without adding noise is quite a task I'd say that searching for low noise op amp is what you should do. A quick googling landed me right on the TI low noise amp page. As you can see there are sub \$\frac{nV}{Hz}\$ amplifiers, which may be a good start.
Please note that if your signal is low pass with a narrow band (such as and EEG) you need to consider also flicker noise and offset. If this is an issue you should really switch to an instrumentation amplifier that includes a chopping modulattor or at least an auto-zero or a correlated double sampling input stage. This first amp can help you give a boost to your signal, somewhere around 40dB, you can then add another stage to add the rest of the gain. Please note that in a multi stage amplifier the first stage is the most important noise wise, i.e. your second stage may be much more noisy than the first.
Best Answer
Here is the thing, noise is actually a misnomer, signal distortion is what it should really be called.
Every time you pass a signal through any component or even a trace of wire, the signal that comes out the other end WILL have some distortion compared to the original signal. Some of that is "filtering" and "reflections" some of it in cross-coupling from other signals outside of the intended signal path (noise).
Voltage Divider
If you can pass your signal with a signal to noise ratio of X through an ideal divide by 10 resistor divider, your signal would still have the same "noise ratio" of X.
Unfortunately, in reality there is no such thing as an ideal anything.
After division the signal will be distorted a little by inductances and capacitances and the physics of the resistor itself. Further, you just built a signal mixer to add in whatever noise is on the ground side. How much distortion is introduced depends on the quality of the circuit and the nature of the signal. The "noise" distortion could be lower but usually it will be higher.
Do It Late
Reducing a signal level is generally something to be avoided. As I already mentioned, you are actually mixing in the ground noise, but you are also producing a signal that is now more SUSSEPTIBLE to the ambient noise in the system. As such, if it is absolutely necessary to reduce a signal to feed into some device, like an ADC, it is prudent to do that reduction as late in the signal processing chain as possible, and physically as close as possible to the ADC.
Amplification
The same goes for amplification. The entering signal along with the noise will again be distorted on the way through the amplifier. Different frequencies will again be distorted by different amounts. We actually design circuits to take advantage of that and call them filters.
As for feeding into an ADC.
ADCs compare an input signal with a reference signal. Obviously, if the reference is noisy you will get LSB comparison errors. If there is an ambient noise level, then that can couple into and produce a larger "noise component" on a low level reference compared to a larger reference. As such, ADCs work better in general at the larger end of their acceptable signal range.
HOWEVER: That does not necessarily mean amplifying the signal so you can maximize the reference of the ADC is the right thing to do. If that same noise is coupled into the signal you are trying to measure before you amplify it, you are back where you started, but now the signal carries the added distortion of the amplification.
Balance
There is a balance in there somewhere. Ultimately, the best method is to limit the number of times you have to mess with the signal on it's way into the ADC and keep the ADC's reference at a level that you can tolerate the reference noise. Less "messing" also limits the effects of component tolerances. And of course, keep the signal and reference as "quiet" as you can. Some circuit tuning is often required to optimize the ADC chain.
Cost
Cost can often be also be a limiting factor. More accuracy generally means more cost. Part of the design process also involves deciding how much error you can tolerate and how much extra cost you can afford to get there.