In Pride and Prejudice, Mr Darcy spends about 3 weeks visiting Lady Catherine De Burgh with Col. Fitzwilliam, who was a great friend of his mother's.
Mr Darcy is a man of vast independent means, and it is made clear that while Lady De Burgh wants him to marry her daughter, he has absolutely no interest in the match. It is also made clear that he has a very low opinion of Lady De Burgh herself.
To a modern reader, it seems odd that he should spend such a long time visiting someone for whom he has no personal affection, nor any expectation of material advancement: Lady De Burgh can give him absolutely nothing which he needs or wants. And I don't think it is suggested in the text that he contrived the visit simply because Ms Bennett would be there, or that he was even aware of that fact?
Presumably to a contemporary reader, the visit is not an odd one though. Would visiting family friends for this length of time have been common for someone of Mr Darcy's class? Why?
Edit: Mods I'm sorry I added the wrong flair and now seem unable to change it myself.
One of the hardest things for a modern westerner, particularly a modern American (I am also one) to grasp about eighteenth- and nineteenth-century life is the amount to which duty, obligation, propriety, etc. played into aristocratic and gentry lives. We tend to have a cynical view of rich people as only doing what they find/found pleasant and/or what could be useful to themselves in a realpolitik sort of way. As you say, Mr. Darcy does not enjoy Lady Catherine's company and he doesn't want to marry Anne, and he can't be keeping her happy in order to get a living from her (like Mr. Collins) or inherit from her (he doesn't need it and she's almost certainly going to keep it in the de Bourgh family anyway). So why?
One thing that is made clear in the text of Pride and Prejudice is that Mr. Darcy believes deeply in his obligations in both positive and negative ways. He tries to keep Jane Bennet away from Mr. Bingley because he sees it as his duty as Bingley's friend to prevent him from being taken advantage of by schemers (as he believes the Bennets to be); he's also a good employer who never says an angry word to his staff and looks after all of his tenants, the ideal and idealized wealthy landowner.
It feels silly to say this, because of course family is important now, but family networks were extremely important to the Georgian gentry. Being on good terms with extended family members was important, both to be genteel and to pass needed information about local matters or happenings within the family. There is a difference between enjoying someone's company and being willing to spend time with them, after all! Lady Catherine is his aunt - both Lady Catherine and his mother were daughters of an earl, sisters of Col. Fitzwilliam's father - and it would not be proper for him to neglect his closest relative outside of his sister. It's his duty to visit her probably once or twice a year, and Georgian gentry visiting meant staying at least a few weeks, in part because travel was more difficult and took longer, and in part because it's not like they had jobs to get back to. It wouldn't be at all odd for him to do it, and to the contemporary viewer it would have shown that he's a good person because he does it even though it's unpleasant. He probably also goes to the Fitzwilliam family seat as well to show respect to his uncle.