I've tried that, and we didn't get far. A list of ideas to throw thier way would go a long way.
In my experience players tend to be a bit scared to actually add new things to the world that their DM is designing or to think of things that could be considered an "adventure". For example, recently we discussed the very same thing with our group and one of the players pointed out that his character was a follower and would not go out of his way to do stuff which made it a challenge for him to fill in the blanks. The DM then pointed out that he was perfectly free to think of NPCs that would hire him to do adventurous things as long as it was something a level 17 PC would have no problem dealing with and that would not automatically generate something that would be considered a game mechanical benefit for a level 17 PC. The DM came with the example of having the PC being hired as a guide in an expedition to a jungle to find valuable mineral deposits (which fitted this PC's background and skills perfectly) or something similar. My own PC in that same campaign has performed his share of off-screen heists some of which that had consequences in the regular sessions and others that never were more than a header in the local newspaper. It was the DM who decided though whether or not their were any long term consequences.
Mind you, the advice is not for every player. It does not only require confidence and trust from both sides of the screen, but also a certain level of knowledge on how the setting operates, creativity and a certain commitment to active involvement in the game. It is also not something that you should do at the game table in the middle of a session, but at the end and then something to discuss in between. It gives the player some time to contemplate the situation. If your players don't show any interest, then let it rest. Personally I also wouldn't bother with thinking of minor events for that PC, since they add little. If you want to though, simply grab your local newspaper (or scan the web) and check for some news that grab your interest whether it is "car crashes into spectators of New Year celebration, wounding 16, killing 1" or "police force reorganized, 1/3rd of the new managers is woman" (to pick two from the paper I read this morning). It tends to be easy to adapt these to your average fantasy setting and give the players a role in it. Just watch for any potential sensivities. You don't want to get your players angry because you used something in what they consider to be a disrespectful way.
As for the players walking all over the DM, if that were the case then the OP would have said something differently then "we didn't get very far" as a reply 
The DM would also have a lot more issues beyond "what can a PC do during downtime", not to mention that in those circumstances the players would have no issue answering that question in the first place.
