Sorry that these aren't more organized, but here goes:
Have a clan tab on the roster page that shows all members of your clan regardless of whether you are already friends with them (so that you can actually add them easily or just join up with them quickly).
Make it so that clanmates sort of show up as friends - i.e. they can join if you're in "friends only" mode without the friend request.
Clan challenges - daily, weekly, monthly - it doesn't matter. It could be something like "while paired up with at least one other clan member, kill x members of rival clans in PVP" or "Have your clan complete x bounties in a month" (100-200 seems reasonable, though even 500 isn't too out there).
Designated (or instanced) social space for your clan.
Leaderboards within a clan (reset monthly or weekly) could also be interesting (could show up at the above mentioned social space).
New PVP option: Clan Wars (could be a monthly thing like Iron Banner) - Could be any of the team modes, but you have to enter with a full fireteam (either 3 for Skirmish/Salvage or 6 for Control/Clash/Combined Arms). Leaderboards at the end with bragging rights on the line (maybe a unique emblem or something similar to the Nightfall glow, but around your entire body for the top 20% of teams - no other bonuses).
10% boost to Iron Banner reputation gains for every member of your clan in your fireteam.
Allow clan admins to invite individuals to the clan directly from the game (and manage the clan, while you're at it).
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I support this