Donna was originally introduced into the Teen Titans as Wonder Girl, Wonder Woman’s sidekick, in Brave and the Bold #60 (June-July 1960). However, Wonder Woman never had a sidekick in her own comic so Donna was a whole invention by the Teen Titans creators Bob Haney and Nick Cardy. Later creators have tried to reconcile Donna’s origin with the Wonder Woman mythology to varying degrees of success.
Donna was the first of the Titans to marry in Tales of the Teen Titans #50 (February 1985) and to have a child in the 1992 “Total Chaos” crossover. She was already estranged from her husband when he and their son were killed in a car crash in Wonder Woman vol 2. #121 (May 1997). Donna had became too old to be called Wonder Girl and had had a couple of different identities. First as Troia (as an avatar of the mythical Titans) and then as a Darkstar (a group set up by the Controllers in order to compete with the Guardian’s Green Lantern Corps). She has since defaulted back to using her own name and does not use a codename.
Donna’s recent history begins with her dead at the hands of a malfunctioning Superman Robot in 2003′s Graduation Day mini-series. This was the series that was the catalyst which saw Young Justice and the Titans reorganise into the Geoff Johns’ Teen Titans and Judd Winick’s Outsiders. It was explained in 2005′s The Return of Donna Troy mini-series that Donna had actually been reborn as one of the Titans of Myth, a group of immortals connected to the Olympian Gods, who were planning on escaping the impending chaos of the Infinite Crisis. The Teen Titans/Outsiders restored Donna’s memory and freed her from the Titans of Myth’s plans. It was explained that Donna had a unique link to the Multiverse because Dark Angel, an evil duplicate of herself from the first Multiverse, had repeatedly interfered in her past. This explains why Donna’s back story so often seems in flux.
Donna played a significant role during the Infinite Crisis when she used the resources of the Titans of Myth to transport a team of heroes to the centre of the Universe and into the heart of the Rann-Thanagar War. This set-up Starfire, Animal Man, and Adam Strange’s journey home in 52 and the discovery of the 52 Multiverse by the Red Tornado. Donna also became the custodian of the Orb of the Monitor with which she bore witness to the “History of the DC Universe” during the backups in 52. Later on, during the year shown in 52, she briefly succeeded her sister as Wonder Woman. The Monitors of the New Multiverse saw her as an anomaly as the believed that she was meant to have died during the Infinite Crisis (a wrinkle in reality mean that Jade died instead). She went on the run from them in Countdown to Final Crisis with Kyle Rayner (Green Lantern and her ex-boyfriend) and Jason Todd (the formerly dead Robin). She eventually met a another corrupted version of herself:

Once she’d returned from space and the Multiverse Donna rejoined the Titans, but she never felt settled. She described the feeling as,
Sometimes I feel like I was never young. I was saving the world before I could drive. We all grew up so fast. Well, most of us, but from the start , I was always the adult. The big sister. The den mother. Always looking after everybody else. Always there for them. So why do I feel so alone.
Donna Troy, Titans #20 (Feb 2010) by Mike Johnson
When the monster Genocide (a Frankenstein-like aberration created by the Secret Society) attacked Wonder Woman Donna stood by her side. Psychic feedback from Genocide played on Donna’s fears and she began blaming Diana for her family’s deaths. She fought Diana on the shores of Paradise Island until wakened from her hallucinations. Unfortunately that imbroiled them in a complex powerstruggle between the Gods, the Amazons, and the newly created Olympians (Zeus’s male Amazons). She stood by Hippolyta and Diana’s side when they faced Zeus and forced him to confess his failures (Wonder Woman #36-39, Nov 09-Feb 10).
Donna had been trying to restart her private life by accepting new photography commissions (she was a photographer back when she was married), but her first job turned out to be a trap set by the Fearsome Five. Nevertheless, she rented a new apartment in Miami and began dating again (Titans #20). Donna was present at Animal Man’s house in Cry For Justice #5 when Congorilla and Starman recruited his help. And it was Donna who captured Prometheus in Cry For Justice #6.
Her live was unturned again when the corpses of her dead son Robert and ex-husband Terry were reanimated as Black Lanterns. She was bitten by one of them and began to sense Nekron’s power even before he turned her into a living Black Lantern (Blackest Night: Titans #1-3, Blackest Night #5). And for the record, that undead baby was the creepiest, sickest thing in the entire run of Blackest Night – not even undead Black Lantern Doctor Light came close. Donna was at the Hospital to retrieve Robert’s corpse for reburial when we first see her in JLA #41.
Why does Donna deserve to be in the Justice League? In one way or another she has been a principal in almost every major plot line from her return through Blackest Night. Right now she’s A-list DCU. Donna is obviously standing in for Wonder Woman, but there is a prescient for this. When Hippolyta was punished by being made Wonder Woman (following Diana’s death) she took her daughter’s place in the Justice League. Also, when Artimes took over the Wonder Woman mantle she tried to replace Diana in Justice League America. In the modern comics Donna is Diana’s twin sister, not her younger sister. She is as fast, as strong, and as tough as Diana and she may even be a more rounded person. So she’s a natural and deserving JLA candidate if and when her sister isn’t available.