Lucie Arnaz, whose illustrious pedigree is evident in her name, and actor Laurence Luckinbill were Simonized several years ago.

He was on Broadway doing Neil Simon's "Chapter Two." She was on Broadway doing Neil Simon's "They're Playing Our Song." They met at Joe Allan's, the famous Broadway restaurant, and started seeing each other entr'acte.

"You never think that those Broadway romances will turn out to be anything," says Arnaz. "I think this one stuck because we weren't each other's leading man or leading lady and so we didn't see each other all the time."

However, they did make a television film together soon after, a typical video love story about a dry cleaner who falls in love with a lady lawyer. It was called "The Mating Season," and right after they finished it, they got married.

Twenty months ago, they had a son, whom they named . . . Simon. The famous playwright was obviously proud to be thus remembered. He repeated the name out loud. "Simon Luckinbill, Simon Luckinbill . . . sounds like a Nazi-chaser."

Right now, Arnaz, 31, and Luckinbill, 47, are sharing a Watergate suite, and they're sharing the star dressing room at the Warner Theatre, where they opened a two-week run of "They're Playing Our Song" last night. "It's the Yul Brynner room," says Luckinbill. "Or the Yul Browner room," a contractual legacy of the run of "The King and I," when Brynner had it painted his favorite color.

The billing, they admit, can be a delicate question. "We try to do equal, left-side/right-side billing whenever possible," Arnaz explains. "It's a stupid thing because whether you're doing television or a Broadway show, you have a different rating. Larry is by far the more distinguished actor in the theatre, and he's probably done more television than I have, but maybe I have a higher TVQ rating because I did a series with Lucille Ball." Lucille Ball, of course, was mom.

Lucie Arnaz is the ultimate Desilu Production. Lucille Ball was producer, Desi Arnaz was director. There were two acts, Lucie Arnaz and Desiderio Alberto Arnaz IV.

Her younger brother may have gotten a little more attention back then; after all, his birth was mimicked on the Jan. 19, 1953, "I Love Lucy," when Lucy Ricardo gave birth to little Ricky Ricardo the same night that Lucille Ball was giving birth to Desi Jr.

Later, little Lucie got to do some guest shots on those early shows, but shied away from regular appearances until she co-starred in the latter-day "Here's Lucy" (1968-1974), which also featured Desi Jr. It was Lucie's first real venture into acting. She was, she says, "reluctant to stay on the same trail" as her mother. "I wasn't stupid enough to turn down the opportunity to work with her when she offered it because it was a tremendous experience. But I didn't think television was where I was going to end up. I had a good time there, but it wasn't anything that was fulfilling me."

In the age of troubled celebrity children getting revenge in print, Lucie is not about to spill any beans on her mother or her father, Desi Arnaz. "People approached me nine years ago," she recalls, still somewhat incredulous. "I was offered lots of money, but I said, 'Shouldn't I wait a little longer till I know a little more?' And I have the worst memory. I do keep a journal, but it's about my life, not hers."

The Ball-Arnaz marriage was occasionally volatile, but "I have no scandal about my mother, that's for sure. She was a good, average mother who tried hard, and my dad was a wonderful father. They both had tempers and they were both strict parents, but I couldn't write anything that would be scintillating or scandalous about either of them. The family was always first, but work was so much fun that I never got the feeling they were doing it just for the money or the star trip.

"They weren't celebrities, they hated that game; they never went to Hollywood parties. I got a sense of acting as a craft--it seemed like a family business. I learned that you had to be professional, you can't fool around in this business. If my brother and I hadn't done well on "Here's Lucy" we would have been out after a year."

Having come to acting somewhat reluctantly, Lucie worked at it conscientiously, concentrating on summer stock and touring companies of shows like "Seesaw." She also slipped into films like "The Black Dahlia" and "Billy Jack Goes to Washington," the Tom Laughlin remake of "Mr. Smith Goes to Washington," which was made in 1976 and disappeared almost immediately amid legal wrangles. The film, which prophetically deals with secrecy behind nuclear reactor projects, has been re-edited and is scheduled for a September release, which is no great news to Arnaz. "I was 30 pounds heavier and God only knows what kind of actress," she moans.

What worries her is that her latest film project, "Second Thoughts," hits the screens one month later. "I'm going to tell people I did a De Niro Robert De Niro, who gained weight intentionally for his role in "Raging Bull" , went to Washington to study senatorial aides and found they got roly-poly from their desk jobs and had kinky hairdos. Then I can tell everyone I went on a crash diet for 'Second Thoughts.' "

She and Luckinbill have worked together several times before this tour of "They're Playing Our Song," including a West Coast production of "Whose Life Is It Anyway?" in which they alternated playing the paralyzed sculptor. Luckinbill, who graduated from Catholic University's drama school, has a staggering list of professional credits, including a Tony Award nomination for his work in the Pulitzer Prize-winning drama "The Shadow Box" (by yet another Catholic grad, Michael Cristofer). The Simon play, however, is his first musical in 18 years. "Nobody could ever get him to sing except at home," Arnaz chides gently.

"When we started out, we thought off all the things we could do together, that we'd be the Lunts. But people thought that's all we wanted to do, and so we had to be careful because then there go the scripts he would normally have gotten before that didn't have a part for me."

So now they're particular, though Arnaz originally accepted "Washington Mistress" thinking he would play the senator (Richard Jordan got the part). She got strong reviews from that TV shot, as well as from her small but luminous role in the Neil Diamond debacle "The Jazz Singer." "I did scenes with Laurence Olivier, probably the most experienced actor in the world, and with Neil Diamond, who'd never done a picture before in his life," she recalls with a laugh. "Let's put it this way--it was a very balancing experience."

There are new projects and some bones to bury as well. On Aug. 31, CBS will show the rejected pilot for "One More Try," a TV series Arnaz and Luckinbill co-produced and co-starred in for Universal. It was "a romantic comedy" about second marriages and stepchildren, a real-life situation for the Luckinbills: This is the second marriage for both, and they have his two children from the previous marriage. CBS called to let them know about the air date, to which Luckinbill sarcastically responded, "Is there an August 31?"

There's an even bigger, long-term project that Luckinbill has been scripting for some time. It's a mini-series about Adah Isaacs Menken, the 19th-century actress who had a blazing craeer and then disappeared. "She created herself, had the guts and strength of mind to do it; she created the tour, the 8-by-10 glossy," Luckinbill says. "She was a phenomenon--a poet, lover of Swinburne, Dickens and Bret Harte; she was married four times, to the heavyweight champion of the world, to a Jewish musician. Hers is a story of a human being trapped in an image that obliterates her, constructing a public image and then being trapped in it."

It's a theatrical phenomeneon that Arnaz and Luckinbill feel occurs every 50 years: It resurfaced with Mabel Normand in the '20s and with Marilyn Monroe in the '60s; all three women died tragically in their 30s. Luckinbill sees Arnaz as Menken, and it's a role she relishes (she's already played Normand in "Mack and Mabel"). It may have to wait, though, because there's another, very important production scheduled for Dec. 31.

This time, they hope it's a girl.