Of American-English origin, the noun Energizer bunny, or Energiser bunny, also with capital initial in the second element, designates a persistent or indefatigable person or phenomenon.

This noun refers to Energizer Bunny™, the name of a battery-operated toy rabbit represented as never running out of energy, featured from 1988 in a television advertising campaign for batteries—as mentioned in a letter by one Cindy Newsome, published in the Detroit Free Press (Detroit, Michigan) of Thursday 1^st December 1988:

My new favorite commercial: The Energizer bunny playing the bass drum. Is he cool or what?

An article describing the television advertising campaign appeared in several U.S. and Canadian newspapers in October 1989—for example in The Vancouver Sun (Vancouver, British Columbia) of Monday the 23^rd:

Right from the start, the new commercial for Eveready’s Energizer batteries does not look like the run-of-the-mill television ad.
The star of the whimsical campaign is a bright pink toy bunny wearing Ray-Ban sunglasses and blue thongs.
Banging a big bass drum as it struts around the set, the mechanical rabbit suddenly runs amok and marches out of the sound studio’s doors.
But unsuspecting viewers watching when the spot made its debut last week were in for an even bigger surprise.
Halfway through what appears to be a coffee commercial that shows two women quietly chatting, the runaway rabbit pops up and creates havoc as it tramps across the table.
Then, just when it seems safe to go back to ignoring commercials, the Energizer bunny charges through the laboratory setting of a spot for a nasal spray.
Startled viewers were left wondering when the marauding rabbit would turn up next.
That is just the reaction Eveready wants.
“Consumers see so much advertising that they are bored with the average commercial and we thought the best way to catch their attention was to have the Energizer bunny seem to be popping up in other people’s commercials,” said Tony Wright, vice-president and management supervisor at Chiat/Day/Mojo, the advertising agency that created the campaign.
[…]
To fight for attention, Eveready has come up with a twist on what are known in the industry as “bookend ads.”
A bookend is a commercial for one product that is split into two parts, separated by commercials from other advertisers.
Well aware that once people catch onto the gimmick they will tune it out, the agency is already developing a new series of 20-second spots in which the wayward rabbit will break into in phoney commercials for other products.

The noun Energizer bunny soon came to be used metaphorically—here are seven early occurrences:

1-: From Bob Talbert’s column, published in The Detroit News and Free Press (Detroit, Michigan) of Sunday 3^rd December 1989:

LOSER: The Detroit Red Wing power play. Even the Energizer Bunny is more aggressive.

2-: From the following by Dave Glenn about Scott Williams, of the North Carolina Tar Heels football team, published in The Daily Tar Heel (Chapel Hill, North Carolina) of Wednesday 7^th February 1990:

When Williams is on from close range, he’s about as unstoppable as an Energizer bunny on amphetamines.

3-: From an article about the Delaware Blue Hens, by Jack Chevalier, published in The News Journal (Wilmington, Delaware) of Tuesday 27^th February 1990:

The Hens have lost their ECC tourney opener for nine straight years. But this club has the defense and athletic ability and, most of all, the adrenaline to reverse that awful trend.
Take freshman guard Jeff Haddock, for example. This little guy with the blank expression appears fairly harmless. But when something goes right, Haddock turns into the Energizer bunny operating at frenzied speed.

4-: From the television column written by Tony Shales, published in several U.S. newspapers in March 1990—for example in the York Daily Record (York, Pennsylvania) of Wednesday the 14^th:

You’re lucky if you can walk down the street without running into the phantom of the opera. This guy is more ubiquitous than the Energizer bunny. Ever since he was unearthed for Andrew Lloyd Webber’s ¹ 1988 Broadway musical, the phantom has been as hard to avoid as the IRS ².

¹ Andrew Lloyd Webber (born 1948) is an English composer and impresario of musical theatre.
² IRS is the abbreviation of Internal Revenue Service.

5-: From an article by Steve Dimeglio about Luis Sandoval, a sophomore distance runner for Coachella Valley High School, published in The Desert Sun (Palm Springs, California) of Tuesday 1^st May 1990:

Like the Energizer bunny, Sandoval keeps going, and going, and going . . .

6-: From the column written by Ann Grauvogl, published in the Argus Leader (Sioux Falls, South Dakota) of Friday 24^th August 1990:

Kind of like the Energizer bunny, Pretty Woman ³ is the movie that keeps on working. The Richard Gere-Julia Roberts love story that’s based on My Fair Lady ⁴ begins its 23^rd week in Sioux Falls today.

³ Pretty Woman (1990) is a U.S. romantic comedy film starring the U.S. actors Richard Gere (born 1949) and Julia Roberts (born 1967).
⁴ My Fair Lady (1956) is a musical based on Pygmalion (1913), a play by the Irish author George Bernard Shaw (1856-1950).

7-: From the account by Dave George of a basketball match between the Miami Heat and the New Jersey Nets, published in The Palm Beach Post (West Palm Beach, Florida) of Thursday 29^th November 1990:

Grant Long was inspired against the Nets, working fiercely to deny Coleman the ball in the second half and scoring a season-high 13 points, but he is a spot player. Kevin Edwards (20 points) and Glen Rice (eight assists) were wound up Wednesday, but Energizer bunnies they ain’t.

The noun is in use in British English, too. For example, the following about Neil Black, of the English rugby-union team that played against South Africa in the World Cup, was published in The Guardian (London and Manchester, England) of Monday 20^th October 2003:

Neil Black
Brilliant in the first half as England came under pressure. An Energiser bunny in wolf’s clothing.