Short Answer: We allow everyone to nominate anyone else for Inspiration, the DM just has to say Yea or Nay on the spot. Quick and easy. We try to use it as often as possible, but have a hard time getting people to remember it's there.
My little brother has become our new standard DM ("little" as in 4 years younger than me, but now that he's 39 years old he isn't actually very little anymore!) and despite just starting his first actual campaign 2 years ago, after 23 years of playing (aside from a short stint running an Al-Qadim campaign for a few months back when he was 16 that was pretty hairy) he has turned out to be an absolutely spectacular Dungeon Master, and he's now additionally running a second regular game for a different group of friends, with me being the only shared member between the two groups. There have been a few little rough spots, but nothing anywhere near major, and he's left me in utter awe of his raw ability as a DM. He's always been very charismatic, to the point of becoming the leader of most of our D&D parties from almost the first time he played, despite him being 15 playing with a bunch of 20+ year olds. He would always just take the initiative and everyone else would follow just because it felt natural, and it worked out perfectly because he loved playing very charismatic Paladins, and did it wonderfully.
He and I try to remind everyone of Inspiration and encourage them to use it, and have worked out a system where anyone in the group can nominate anyone else for Inspiration, which the DM then has to approve on the spot (which he usually does unless it's for a dumb reason.)
That way we hope to get everyone spending and re-earning Inspiration several times a game, with the comparison I like to use being a line I remember from something about the old DC Heroes game, where you could spend your Hero Points both as "experience" to increase your character's abilities, and to modify rolls and avoid taking damage during the game, and I remember reading something about a good exciting game having Hero Points "flowing like water" because they were being used so frequently.
Anyone in the group can nominate anyone else, mostly for good roleplaying, but also possibly for cool ideas or a good joke. We've even also taken to nominating the DM for "Inspiration" but that doesn't actually have any game effect, it's just a way to let him know he's doing a great job and we're truly enjoying his game.
In this way, we're actually using Inspiration as a continuation of something we'd been doing for years already; "Rock 'N Roll Points!" Any time anyone in the group did something cool, something that made everyone else sit up and say "Whoah!" we'd collectively give them a "Rock 'N Roll Point!" just for fun, not really meant to do anything, and we just all sort of started recording our "Rock 'N Roll Point!" totals, until eventually the DM would occasionally allow us to spend one of them to reroll a really bad die roll in extremely important situations. And when 5th Edition first came out and we all read the PHB, we all looked at each other and said "they made actual rules for Rock 'N Roll Points!" So that was pretty Rock 'N Roll to us.
My brother and I really like the Inspiration mechanic as a mechanical way to encourage good roleplaying (as well as liking the larger Advantage mechanic itself) and do everything we can to encourage everyone to take advantage of its existence, but it's hard to get everyone to remember, but that usually just boils down to me spending my Inspiration as often as I can, and him granting Inspiration to people as often as is warranted, but that frequently doesn't mean anything because they still have Inspiration that they haven't used yet. So it's kind of hard.