Some researcher out at MTM Enterprises is in a lot of trouble. The new show "Mary," starring Mary Tyler Moore, whose television career is moving from television to newspaper reporting at the fictional Chicago Post, is upsetting the owner of a newspaper in the Windy City called the Chicago Post. With the debut of Moore's new television series next week, the publisher of the real Chicago Post is trying to stop the use of his paper's name.

Chicago Alderman Richard Mell, who is publisher of the Post, a free monthly publication aimed at his Northwest neighborhood, says his newspaper is protected by federal and state trademark laws. Mell is threatening to make a big thing of it, saying he will go to court if the show's production doesn't "cease and desist."

"What if I used their cat logo MTM Productions symbol as a dog food producer," Mell said. In the show, Moore plays an advice columnist in a casual newsroom surrounded by crazy characters. Mell's attorney has been in contact with MTM's lawyer, but he feels he was not taken seriously. A small sum of money -- which Mell wouldn't specify -- was offered to clear up the matter but he refused it. John Eaton, Showcased

Washington's premier jazz pianist, John Eaton, is finally going to get the kind of spotlighting he deserves. A man of tremendous talent and a sophisticated touch, he has often had to play against offensive, noisy drinkers at clubs around town including the Snuggery and the Ritz-Carlton and at the Embassy Row, where he now performs in the Wintergarden Room. That's about to change. Instead of being almost used as background music, the Embassy Row will now showcase Eaton in scheduled nightly performances.

Like other great nightclub performers such as Bobby Short at the Carlyle Hotel in New York City, Eaton will have shows scheduled Tuesday through Thursday at 9 and 11:30 p.m., Friday and Saturday at 9, 11 and 12:30. Embassy Row general manager Gabor Olah said, "The Eaton show is now professionally staged and is acoustically perfect for the Wintergarden."

And this Sunday, Eaton will have an autograph party for a new recording -- "John Eaton and Friends" -- he cut with Wally Garner on clarinet, Tom Cecil on bass and Linda Cordray and Alessandra Marc's vocals. They will all be there in the early evening to sign record jackets along with syndicated cartoonist Pat Oliphant, who immortalized the group for the jacket cover. End Notes

There should be a gathering of some State and Defense Department types tonight at the book-signing party for Pamela Sanders' book "Iceland," with photographs by the late Roloff Beny. Sanders lived in Iceland for four years as the wife of the U.S. ambassador to Iceland Marshall Brement, now head of strategic studies for the Department of Navy War College at Newport, R.I. The party at the Madison Hotel is being hosted by Madison owner Marshall B. Coyne and attorney Steve Martindale . . .

Washington actor Bill McLinn, who has portrayed Mark Twain nearly as much as Hal Holbrook, will be in Twain's Hannibal, Mo., today appearing as Twain at the presentation of the Mark Twain/Halley's Comet Aerogram established by the U.S. Postal Service. McLinn, who has appeared as Twain at the Smithsonian, was instrumental in getting the postal service to issue the aerogram . . .

Next week is Christmas party time at the White House, and the thousands of invited guests who will eat, drink and shake hands with the first family will view a great tree in the Blue Room that will be furnished by North Carolinian Hal Johnson, the National Christmas Tree Association's grand champion grower. Nancy Reagan will receive the gift tree today. Johnson is also furnishing the tree in the family quarters, as well as about a dozen others for the Executive Mansion . . .

Rambo may be big time in the United States, but the British Broadcasting Corporation has canceled plans to show the blood-and-guts film as part of its Christmas schedule. The announcement came three days after Prime Minister Margaret Thatcher said the government was considering action to curb the amount of violence and sex on British TV.