When presenter/musician/comic Richard Stilgoe had his own show, he would often begin it with an anagram. Anagrams of his own name include "Ricardo Hotlegs" and "Giscard O'Hitler".
The BBC show The Adventure Game is packed with significant anagrams of "Dragon". The residents of Arg, the planet on which the series is set, are called Argonds, and their monarch is the Rangdo (for whom the correct greeting is "Gronda, gronda"). The other Argonds include Darong, Gandor, Gnoard, Dorgan, Ron Gad, Nagord, and the Dogran (a doglike creature). And the local currency, which lends its name to one of the more frequently played games on the series, is drogna.
Blindspot has episode titles that seemingly make no sense but anagram into different words, creating a message that makes sense once you put the 'solved' titles of an entire arc's episodes together.
Castelo Rá-Tim-Bum, Zula is a blue girl. Her name is an anagram of "Azul" (Portuguese for blue).
One toss between The Daily Show and The Colbert Report had Stephen Colbert looking for anagrams in the names of political figures. Since the names were (mostly) real, rather than made up by writers, the results were less than impressive.
On the other hand, Stephen Colbert is very close to being an anagram of Charles Noblet. (Before you try: it has an extraneous A and superfluous L while lacking a P and one more T and E. You end up with STELHEN COLBAR.)
In another episode, Stephen rearranged the letters in CIA Director Leon Panetta's name and came up with Ayman Al-Zawahiri.
Colbert and Stewart also had fun trying to make anagrams out of the name of Reince Prebus, the new head of the Republican party. They ended up with... Prince Reebus? Another notable example was "Pubic Re-rinse".
On Coronation Street, a character named Carter sets up a fake company to defraud his boss, Mike Baldwin... and, rather foolishly, names said company "Artrec". After catching him out, Mike even points out how stupid it was to use an anagram name.
Criminal Minds does this at least twice, both times a little strangely: once, when a man who thinks he's the Fisher King uses the alias "Sir Kneighf" to hide his (delusional) identity, and once when a serial killer known as "the Reaper" goes into hiding under the name "Peter Rhea" (intentionally, so as to make it possible for the team to find him). Reid, of course, figures out both anagrams in record time.
"Genesis of the Daleks": The Kaled race are the precursers of the Daleks. The Doctor even lampshades it "K-A-L-E-D, that's an anagram of... How interesting."
"The Leisure Hive": The Foamasi, an alien society that has criminal organisations called "Lodges", have a name that is an anagram of Mafiosa.
"Meglos": One of the villains is named Brotadac, an anagram of "bad actor". The writer's assumption was that they'd hire a bad actor for the part, on account of it being a fairly mechanical Stock Character role. The part was eventually given to a veteran character actor, who fortunately found the anagram hilarious when someone pointed it out to him.
In "The Keeper of Traken", the Master possesses the body of Tremas. In the later story "The King's Demons", he disguises himself as Sir Giles Estram. Out-of-universe, Significant Anagrams were used to hide Anthony Ainley's name in the credits and in Radio Times cast lists when the Master was in disguise, including "Neil Toynay" ("Castrovalva"), "Leon Ny Tai" ("Time-Flight"), and "James Stoker" (from "Master's Joke", in "The King's Demons"). The joke even extends to Big Finish's Doctor Who audios, in which "Sam Kisgart" (from Mark Gatiss) is credited as the actor for the Unbound Master.
"The Two Doctors" featured a race called Androgums, who were obsessed with eating and governed by their appetites. "Androgum" is an anagram of "Gourmand".
"Remembrance of the Daleks": Terry Molloy was credited under the name "Roy Tromelly" as the Imperial Dalek Emperor so no-one would realise that the Emperor was really Davros.
In the Big Finish audio "Zagreus", neither Eight nor Charley get "Saviltride" until it's spelled out for them: "evil TARDIS".
There's Torchwood, which is an anagram of Doctor Who. Before the name was introduced in-universe, it was the codename for the revived series during filming.
In season 3 of the revival, the Master runs for Prime Minister using the alias "Mister Saxon" an anagram of "Master No. Six" John Simm being the sixth incarnation of the Master to appear in the series. However, according to Russell T Davies, this is just a coincidence: once you've used "Mister", it's harder to avoid a "Master" anagram than to achieve one. It also does require you to both write a word in full when it's normally abbreviated (Mr./Mister) and abbreviate a word when it's normally written in full (number/no.).
Donna Noble is actually an anagram of "a non blonde", a reference to the new series' first companion, Rose Tyler, and the various blonde jokes that her successor Martha Jones quipped during season three.
In the same "coincidence" line, Wilfred Mott anagrams to "WTF Time Lord" or "Time Lord FTW".
Also in "Journey's End", the "Osterhagen" in "Osterhagen Key" is an anagram of "Earth's gone".
Amy Pond's actress Karen Gillan revealed in an interview that her auditions were conducted for a project named "Panic Moon", which is an anagram of companion. Producer Steven Moffat, who was also present, then joked that he would have to find a new anagram for future companion auditions.
The anagram that wasn't significant: Astrid Peth is an anagram of TARDIS, with peth being Welsh for "part". That's good, since the placement of a P in "The TARDIS" wasn't exactly obvious. Oddly, she's an ordinary Human Alien with no connection to the TARDIS. Although in the end, her spirit (represented as gaseous stardust not unlike the way the TARDISs heart is often depicted in the series) is released into space and allowed to travel the universe adrift like she always wanted, kind of like the TARDIS itself.
On Dollhouse, Alpha uses the alias E. Hap Lasher, which is an anagram for "Alpha's Here".
In the pilot episode of Ellery Queen, the names of the fashion designer/victim's clothing lines are anagrams of the names of her lovers.
In Fear and Faith, one of Derren Brown's many documentaries about human psychology, the placebo used the name Rumyodin. In The Reveal, he reveals that the name is an anagram of Your Mind.
Though it's never brought up in Garth Marenghis Darkplace, it's been observed that "Garth Marenghi" is an anagram of "Argh, Nightmare."
In the Grimm episode "Nameless", the Fuchsteufelwild, who has been writing "Guess My Name" at each crime scene, is named Trinket Lipslums, which is an anagram for Rumpelstiltskin. Naming by that specific anagram is part of the profile for this sort of Wesen.
The voice of the Death of Rats in the Sky adaptation of Hogfather is credited to Dorckey Hellmice — an anagram of Michelle Dockery, who played Susan.
In the midlife crisis episode of Home Improvement, wise neighbor Wilson points out that 'Tim Taylor' anagrams to 'Mortality'.
The dog Wilson and his (first) wife had was named by her as Hector. She eventually reveals the reasoning behind it: The dog was impossible to house train, and "Hector does go rug" was an anagram of "Doctor Greg House".
House, being House, immediately comes up with a better anagram to use next time: "Gregory House" -> "Huge ego, sorry".
Played for Laughs in the Jonathan Creek episode "The Sinner and the Sandman": a parish community hall has been named after its benefactor, Leonard Corbyn, with an Establishing Shot of the name in large metal letters. When Corbyn gets implicated in a sex scandal, the chairman of the parish council spends some time coming up with entirely fictional benefactors whose names are anagrams of "Leonard Corbyn", because the metal letters were expensive. After the vicar's vetoed making a benefactor up, a later Establishing Shot shows the sign now reads "Ye Old Corn Barn Community Hall".
Kamen Rider Fourze loves this trope like no tomorrow. The protagonists have names based off of past Kamen Riders, while members of the high-level Zodiarts usually have names related to the Zodiac sign they represent.
Yuki Jojima, the Genki Girl. At first it seems that her name pun is simple: Yuki Joji, aka Riderman, as all of the main cast have names that are nods to old-school Riders. However, there's a sneakier anagrammy one: I am Gemini, relating to her stint as the Gemini Zodiarts.
Knowledge Broker JK's full name is eventually revealed to be "Kaizo Jingu", which is an anagram of "Jin" and "Kaizorg", Kamen Rider X's surname and the type of Cyborg he's said to be.
Emoto's anagram is Otome, which is Japanese for virgin. As in Virgo.
Sensei-chan Sarina Sonoda's name can be rearranged into "Sasori nanoda", which is Japanese for "I am the Scorpion", referring to her role as the Scorpion Zodiarts, one of the Big Bad's Co-Dragons.
Erin Suda's name is an anagram of Eridanus, a Greek river that is associated with Aquarius, her alter ego.
In Kamen Rider Ex-Aid, Poppy Pipopapo uses the alias Karino Asuna when in her human disguise as a nurse. In episode 12, Hiiro points out that this is an anagram of "Kari no Naasu", meaning "Pretending to be a nurse". (It's one of the reasons why, to Poppy's disappointment, nobody was the least bit surprised as to her being a Bugster, one of the villain race. If her abilities - definitely not human though largely played for humor - didn't make it obvious, the name did.)
In The Lone Gunmen, all the names used by the character Yves Adele Harlow (including that one) are anagrams of Lee Harvey Oswald.
Lost has had about five meaningful anagrams: Ethan Rom is "Other Man", Mittelos Bioscience is "Lost Time", Gary Troup is "Purgatory", "Henry Gale, Minnesota" = "See Another Man Lying", and the Hoffs/Drawlar Funeral Parlor is "Flashforward". Apparently those have caused many Lost fanatics to begin looking for anagrams in the names of everyone and everything.
Midsomer Murders: In "Blood Wedding", the cryptic inscription 'Cast no sin here' Barnaby finds on the back of an old photo, is actually an anagram for Catherine's son.
In the remake of Randall and Hopkirk (Deceased), another character drops the title and points out that it's a perfect anagram for "sad plonker and real dickhead". The anagram was apparently invented by a critic and incorporated by the writers.
In Reaper, Sam Oliver is an anagram for "Evil Roams".
Parodied in Scream Queens: The Chanels are looking for anything that would point toward Zayday Williams being the Red Devil killer. Chanel #5's evidence is that Zayday's name can be rearranged to say "I MAY SLAY LIZ DAW." Chanel finds this stupid and asks who Liz Daw is, prompting #5 to tearfully say that she doesn't know, but is sure that Zayday is "contemplating on slaying her." For the record, there is no character by the name of Liz Daw.
In Twin Peaks, Windom Earl is an anagram of "Owl in Dream".
On the martial arts show WMAC Masters all the Masters go by their real names and nicknames except one. Tracer, a mysterious man with a military gimmick, is really a kickboxer named Michael Foley who goes by the name Tracy Swedom on the show. It's soon revealed that he's a mole working for the evil cult Jukido. The reason for the fake name is revealed when the Wizard (who's an ex-member of Jukido and aware that his brother Warlock has recently made a secret HeelFace Turn to join them) brings in an anagram generator and enters other masters' names in it, revealing that "Tracy Swedom" is a perfect anagram for "Destroy WMAC".