The question which noise sources need to be taken into consideration depends on how severe they are. Your question indicates that you are interested in noise generated at the op amp and not noise generated by interference from neighboring circuits (internal/external noise).
In order to make things comparable, all noise is referred to the op amp's input (RTI). In theory, I guess any point in your circuit might work as long as you refer all noise sources to that point, but it is common practice to act as if all noise sources were directly at the input pins. Sources include noise in the resistors, noise generated by current flowing into the op amp's input pins and noise that may be considered as a voltage between the inputs pins.
There is a very good discussion at this Q&A-style source and also in this nice article from 1969 (!), both authored by Analog Devices' staff.
Without re-typing everything in these sources, here are some rules of thumb:
Noise in the resistors becomes bad when the resistor values are high (some 100k or some 1M) and when the circuits are designed for high bandwidth since the noise is proportional to \$ \sqrt{4k \cdot T \cdot B \cdot R}.\$
You can try to minimize R, you can try to limit the bandwidth B if possible, you can put the circuit in liquid nitrogen (low temperature T), but you can't go for a low Boltzmann constant, because Boltzmann is dead (quote stolen at Analog Devices).
Current noise, i.e. noise generated by current flowing into the op amp inputs, will be converted to a noise voltage by the resistors around the input (\$R_f\$, \$R_g\$) and amplified by the circuit's gain. This is one of the reasons why one prefers op amps with very low input currents especially for high-ohmic circuits.
Voltage noise results from a real op amp's inability to completely null the voltage between the input pins.
All noise sources can be combined as the square root of the sum of their squares since they are independent of each other, which will work only if all sources are RTI.
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.
Best Answer
to determine the broadband noise figure, which I've learned to simply describe as RNOISE, while not being confused by the low-frequency charges-bouncing-from-traps that produces 1/F random noise, you need to measure the RMS output from DC to about 1 decade above the 1/F frequency corner.
Thus the upper bandwidth will vary, because 1/F corners vary.
Assuming the 1/F corner is 100 Hertz, then set the bandwidth to 1,000 Hertz.
ground the Vin(+).
install series RC from Vin(-) to Ground. Use an R of 100 ohms. Use capacitor suitable to set a high-pass-corner of 100Hz or so; why not. Exact cap is not critical, because 1,000 - 100 is about the same as 1,000 - 0, when expressed as dB.
install feedback resistor for gain of 100x: 10,000 ohms is fine.
add capacitor in parallel with 10,000 ohm, to set bandwidth to 1,000 Hz or so.
Now you have low-noise-contribution amplifier, and you are ready to measure the output random noise.
What noise voltage to expect?
A noisy opamp (typical opamp) will have RNOISE of 10,000 ohms, which is 12 nanoVolts rms / rootHertz.
In 1,000Hz bandwidth (actually 1,000 * pi/2 bandwidth), the input noise will be
The output Vnoise (we have a gain of 100X, remember) will be 360 nanovolts * gain, or 0.36 microvolts rms * 100x == 36 microVolts rms.
Some spectrum analyzers will display this. And a 7A22 scope plugin will display this, particularly with bandwidth set to 3KHz or 1KHz.
You may need another gain-of-100X amplifier, before meters will be usable.
What about Noise Figure?
Noise figure requires some noise density be defined.
The noise figure of this circuit is ----- set by the 100 ohm resistor (in the series RC DC_blocking network, to remove affect of opamp V_input_offset).
Minimum noise of the circuit will be 4nanovolts * sqrt (100 ohm / 1Kohm ohm) scaled by sqrt(bandwidth, scaled by gain.
Thus minimum will be 4nv * 0.31 * 31 * 100 = 4 microVolts.
Plus effect of pi / 2.