Yes, source emitting at the whole solid angle.
I need good convincing-sounding technical terminology that you might hear between a couple radio astronomers, though perfect fidelity isn't necessary. If the question's too broad, think of it this way: I would like it to be a signal which would draw attention as something very weird, but which would not bear any unmistakable marks of artificial origin.
This might be too stupid and broad, and if so I apologize.
I think this is a rather good question. It is not easy to answer though. Here's some thoughts, maybe it gives some body else ideas.
I had to browse through some faint distant memories of how does a radio telescope actually see something. The antenna collects radiation, and the signal is then amplified and processed, either by analogical or digital means. The processing can either use amplitude modulation or frequency modulation, and both have their own acceptable bandwidths; the filtering can be done digitally or analogically, and they remove the higher or lower frequency components from the signal - while the antennae length will also do some preselecting. I would guess that there are multiple stages in the signal processing, where the signal is modulated and multiplexed to different frequencies. The above is valid for a single measurement, the bandpass filters would likely kill the signals outside the frequency spectrum. This means that the telescope is not able to detect signals that are outside it's operating bandwidth, they don't exist in the measurement result. But this doesn't limit measurement results that are collected over time, there could still be some signals with much lower frequencies with sufficient amplitudes can be seen there (more on this later) that work by some other mechanism that is visible within this wavelength range.
I would expect radio telescope beam widths to be comparatively narrow, which means that the source that emits the signal should be within the telescope field of view. There is a possibility of the signal of entering the telescope through some side lobes (it's hard to get rid of these in radio frequencies), but the signal strength would likely need to be rather high, which would mean that there is a high probability that the source would then be directly in the field of view of some other telescope. It would be seen as rather bright, and would likely lead to quick detection. I then thought that if those telescopes were ground based, atmosphere could distort the signal partially. Unfortunately this doesn't happen (much) with radio waves. If the signal suddenly disappeared in certain wavelength, that would likely be detected immediately, but it would be very difficult to say what was wrong. The first thing those poor guys would do would be to check, verify and recalibrate the instrument. In order to make signal disappear or considerably weaken it by a destructive interference; it would require some knowledge of the signals that the telescope sees if it weren't interfered - or pure coincidence, but that's far fetching.
Then, (you might want to verify this), I have a faint memory that there might exist some periodical signals coming from the sun within certain intervals. At least I recall seeing this in incoherent scattering radar measurement data, though it never became clear to me if this was just some coincidence for that particular time, or whether there really are this kind of cycles. Nor did the source of such a signal ever become apparent to me, but I didn't investigate much further than reporting it to the assistant back then. But if such cycles really exist, could the signal slightly alter it, or be very close in frequency and in phase so that it would take time to see those two signals to go out of phase?
Sources for signals that the radio telescopes see should be rather large (in order of metres), the wavelength of the radiation depends on the size of the source. I was thinking of a charged comet, or a comet that has an antennae in it with a weak transmitting power? Or there could be polarization modulation in the signal, I don't know whether polarization modulation is detectable by current instruments. The instrument would likely then see a new signal, and decrease of the new signal when the moving object is outside the FOV. Finding the source of the signal again could be difficult, if it was small and moving.
EDIT: Redefinition of some parts
EDIT^2: Added some details in the text
EDIT^3: Crap, this starts to take my sleeping time. Moving faint object needs some additional data to be written there.
