The four girls said they came up with the new name Little Mix after talking with their mentor Tulisa Contostavlos.
"We decided to change our name to Little Mix as we do not want to cause problems for the charity," they said.
The charity gave a "massive thank you" to supporters who backed its campaign against X Factor.
Little Mix said they had no idea there was a charity with the same name when they came up with Rhythmix.
"We're happy that this has now been resolved," they added.
"We're the same girls as we've always been and we want to thank all of our fans for their continued support.
"We came up with some ideas for our new name which we discussed with Tulisa and she had some ideas too.
"We all agreed on Little Mix as it just felt right."
The music charity Rhythmix, which has been operating in Kent, Surrey and Sussex for 10 years, feared X Factor's use of the name would cause confusion.
It hired lawyers and wrote an open letter to show supremo Simon Cowell asking him to intervene after the mix-up was revealed.
It formally objected to the X Factor's application to register Rhythmix as a trademark in Europe.
Supporters set up a Facebook group urging music fans to buy Nirvana's 1991 track Smells Like Teen Spirit in time to make it the Christmas No 1.
Tens of thousands of people joined the group, pledging to help stop an X Factor act getting the Christmas top spot.
The charity, which receives National Lottery funding, has 50 musicians who have worked with 40,000 young people through youth support services.
Chief executive Mark Davyd said it was delighted the band had changed its name.
"We are very pleased that the girls have, as we understand it, really pushed the programme to allow the name change," he added.
"I think that says a lot about the way they behaved throughout.
"We are very, very pleased with the outcome."
The X Factor girl group Rhythmix has changed its name following a threat of legal action by a Brighton charity with the same name.