Awarding a badge for offering bounties is a great way to promote this feature and help make users aware of it, as well as to encourage them to use this feature to help promote great questions.
However, I'm not 100% convinced that the name of the badge fits the actions, considering we already have existing badges named after peace officers and law enforcement that are awarded for flagging unwanted content.
As far as badge names go, I really like Plutocrat as a name for the badge. A plutocrat is:
someone who rules by virtue of his or her wealth.
Does a huge bounty not determine what questions get the most attention in the StackExchange network? Is this not a form of ruling by virtue of one's reputational wealth?
I also like Tycoon:
person who has a lot of money, power.
Below are reasons why "Sheriff" may not be the best name for a bounty badge:
While I can definitely see the correlation between "Sheriff" badge and offering a "bounty", I feel like that could be confused with the two badges that are awarded to those who raise helpful flags.
Currently, the Citizen Patrol badge is awarded to a user who, for the first time, flags a post. The Deputy badge is awarded for raising 80 helpful flags, and the Marshal badge is awarded for raising 500 helpful flags. When we look at these three badges, it's clear that we're comparing apples to apples but in varying quantities.
The proposed Sheriff badge would likely be confused with other badges intended to recognize volunteers who help keep the site clean, free of spam, and who alert diamond moderators of potential problems. Awarding bounties, on the other hand, is more closely related to awarding the accepted answer than it is to blowing the whistle on a spammer.
Since we already have some badges for those who offer and award bounties, we should look to those existing badges when deriving inspiration for naming a new badge.
The current badges awarded for bounties are the Altruist, Benefactor, Investor, and Promoter, with the first two badges being awarded for awarding bounties and the latter two badges being awarded for offering bounties.
Therefore, with this logic, it stands to reason that a question about a badge awarded to individuals who offer bounties over time should have a name that fits within the same categories as other existing badges related to offering bounties on questions.
Investor and Promoter are those two existing badges related to bounty offerings. Here are the synonyms for those:
From Thesaurus.com on investor:
Synonyms: backer, banker, capitalist, lender, shareholder, stockholder, venture capitalist
From Thesaurus.com on promoter:
Synonyms: advertiser, advocate, ally, backer, booster, endorser, follower, organizer, publicist, sponsor
Tycoon and Plutocrat were words that are more closely related to investor, and are words I found on the same page. These are not names we've used before on badges. However, I noticed that promoter has two synonyms for two badges that we award for sharing links to questions, the Booster badge and the elusive Publicist badge. Therefore, my logic may be somewhat flawed in ruling out "Sheriff" as the name for the badge.
Still, if we are going to promote the creation of this bounty badge, I'd be interested to see a unique name chosen that helps separate it from other types of badges that are awarded for unrelated activities and that emphasize wealth or sharing of wealth.