I have no idea what to do with it. Somehow, I don't think it will fit in alongside my 10/22s.
This makes it sound like it's a gun you'd never have bought even if it hadn't had the grim history attached. In that case, you might want to go towards sale or trade. If you sold it or traded it, I would consider it my duty to inform the buyer (privately) of the history before they were legally committed to the sale. They might not want a gun that'd been used in a suicide either.
Ask yourself also, "If the relative had died of something else, ANYTHING else, would I still want or keep the gun?" If the answer to that question is no, again, getting rid of it may be the preferable option.
If the suicide aspect makes you feel nervous about passing it on, consider offering it to a gunsmith for reduction to spare parts. Only if it's really spooking the hell out of you would I advise destruction.
If the relative meant a lot to you, and you have fond memories of shooting together, that might act as a spur to keep it and shoot it in their memory - but this is a thing only you can decide. I've heard of people shooting a tiny pinch of their loved ones' ashes in their memory, in a gun they used to own, pressed into a hollow point bullet or mixed (in small volume) in with the powder (if reloading or muzzle loading are options). You might consider a gesture like that if you felt it appropriate, before sending the gun on its way.