You have plenty of choices!
I am specialized in X-ray astronomy, then I can suggest you many facilities: they usually give at least some of their data publicly available. Tipically, you have data for light curves (photon count rates versus time), and/or spectra (that is flux versus energy).
I suggest you to work on the light curves: they are easily available, already reduced (you don't have to play with the data to make them usable, you can directly use them), and they fit your request of conforming to a regression model. Also spectra are enjoying, still I know some theoreticians who are not familiar with spectra, plus they usually need some more knowledge, because of response files, spectral fitting packages, etc. It is up to you (and your available time)!
Here a list of some facilities with available data:
Also here you can find a list of many facilities with NASA participation, so the list is much larger.
Of course you can also use other kind of data (optical, infrared, radio) by the MAST project, but I have no experience on those. I guess, in the optical band, the Hubble Space Telescope will have a huge amount of data!
If you need a hand, beyond the SE vote up/vote down neurotic fashion, let me know. Good luck!
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