On Earth-616, Thor and Rogue fought in Avengers Annual #10. She absorbed his physical powers but did not try to pick up his hammer. But having his physical abilities would, at that moment, certainly would not have made her worthy.
This was her first meeting with the Avengers and she was trashing them because of her affiliation with the Brotherhood of Evil Mutants. It would be unlikely she would have been able to lift it under those conditions.
Rogue was able to steal his physical strength and stamina, that was certain. She stands up to a direct venom blast from Spiderwoman, knocks Wonder Man on his butt and holds her own against the Vision and Hawkeye. She had already taken out Captain America and was channeling his power too. It has never been clearly established what the criteria for her power absorption is, or what its limits are.
If she were to engage in a prolonged contact (as she did with Ms. Marvel) it is conceivable she would possess any power Thor had that did not involve Mjolnir to invoke including his storm-summoning, his incredible regenerative and recuperative powers, and perhaps any powers he gains from being the son of Odin and Jord (Gaia). If her power included any memory or training, she would also be able to use his thousands of years of combat expertise as well.
Caveat: Despite the fact Marvel depicts Rogue absorbing Thor's powers/lifeforce in What If? Vol. 2, Issue 66 and being able to wield Mjolnir, in the canon Earth-616, Rogue has never shown the character, nobility, or level of innate goodness that would give her the power to use Mjolnir. As the joke goes, the users of Mjolnir would be a gathering of the sons of Bor and Captain America; a pretty unique group to be sure.
