(written from a Production point of view)
Though an embedded auxiliary craft, known as a "captain's yacht", was already envisioned by Andrew Probert when he designed the Galaxy-class studio model, it was, due to various reasons, never shown in the Star Trek: The Next Generation television shows. However, the final draft of 12 March 1998 of the script for Star Trek: Insurrection explicitly called for an appearance of this kind of craft, at first in scene 136, "EXT. SPACE - THE YACHT - (OPTICAL) falls away from the belly of the mother ship... and after a beat, the thrusters ignite and off they go...",  and a fully-realized vessel and deployment sequence of this kind was finally developed for this occasion. The, what eventually became, Cousteau, was named by Michael Okuda after the vessel named for the esteemed 20th century oceanographer Jacques Cousteau,  who, in turn, acted upon a request of Patrick Stewart to do so. (Cinefex, issue 77, p. 72)
The design of the captain's yacht fell to Illustrator John Eaves, who commented, "Luckily, I had to design only the top of the captain's yacht, because the bottom of it had already been established on the Enterprise itself; there is even a separation line on the miniature studio model. The yacht is a diplomatic cruising vessel that's docked out for touring. It's almost 110 feet long, so it's a pretty big ship." (The Secrets of Star Trek: Insurrection, p. 39) Elaborating he has stated, "It was more of an accident that the shape of the yacht worked out really well, based on that shape down there [on the saucer]. What was good is that the torpedo launcher on the bottom of the Enterprise has a real nice cut line on it. We never thought of it at the time, but it worked out really well as the separation line for the yacht. So, with a few little line changes, we had the bottom of the yacht down. We had the separation from the line of the torpedo launcher right in the middle of that little tower section, and we could separate the yacht and still have that launcher as part of the ship. I kind of made it a touring vehicle – it's meant to be a very luxurious vehicle. We designed it after a real yacht, so it really has a ship look to it." (Star Trek: The Magazine Volume 1, Issue 4, p. 29) He added, "I designed the engine nacelles to fold up into the starship's hull for storage – like the wings on a jet aboard a carrier – then deploy down as the yacht undocks. Fortunately, the detail already present on the starship worked perfectly to accommodating these nacelles." (Cinefex, issue 77, p. 72)
Eaves originally had his design with a recessed windshield, allowing it to dock snugly right behind the Enterprise's protruding torpedo launchers. Yet, as the new ship was in the process of being translated into CGI, the producers hit upon the notion that it would be more aesthetically pleasing to smooth out the front area into the rounded surface as seen, raising the question, according to Eaves (who faithfully complied), "How do they park this thing?" (The Secrets of Star Trek: Insurrection, p. 43) According to his orographic design sketches, Eaves submitted his final designs in February 1998.
Designing the Cousteau's deploymentEdit
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As for the Cousteau's deployment, Eaves remarked," First thing we did with the Federation ships was that there was a captain's yacht, and what we did with that was that we took the bottom of the Enterprise model and logically it was gonna be the little torpedo launcher. So primarily Star Trek: First Contact. And by accident the seam line around the drawing made a perfect without disturbing the torpedo launcher. And so, we took a casting from the bottom of the ILM ship, cut it along that line and built a yacht on top of it, basically. And used a kind of a schooner design to it with the traditional nacelles to it. And so, there is two flanking designs to it, that John Goodson put on the miniature, and they worked excellent on the struts. We did just did a hinge-motion so that the nacelles could drop out, and the whole ship would detach from the whole ship." (Star Trek: Insurrection (Special Edition DVD), special feature, "The Art of Insurrection") Eaves further added,
"Well I'll have to tell you more about the lucky accident of this working out the way it did!!! I had just started on DS9 when we started working on First Contact and I had never seen an episode of Voyager and only two or three episodes of TNG so I didn't even know that a captains yacht was a new design element to Federation Starships. The Torpedo Launcher on the E was basically a new shape for the dome seen on the previous Enterprises and it just made for a perfect place for a main launcher, My Yacht education didn't come till we started to work on Insurrection and and looking at was on the D and Voyager it only made sense to try and manipulate that launcher area into the Yacht, The lines worked out perfectly so as much as it looks like it was intentional it was incredible luck with how the lines worked and how easy it was to make a Yacht bay out of the lower section of the launcher. had I known more earlier, the design would have happened on FC so I'm the one to blame for drafting something in after the fact." 
According to his orographic design sketches, Eaves submitted his final deployment designs in December 1997. It is the deployment of the Cousteau in Insurrection, that has made every other such conceived happenstance superfluous. Robert Bonchune and Rick Sternbach vied to to do a similar thing with the Aeroshuttle, but were nixed by the producers, or as Bonchune has put it, "In a nutshell, back at Foundation, we got into our heads that it would be cool to see the "captain's yacht" of the USS Voyager, that being the AeroShuttle. Rick Sternbach gratuitously did a prelim design and I used part of that and designed the ship you see here. Mojo and I did a whole launch sequence, on spec, meaning "free" and then had it shown to Rick Berman. The response: Mr. Berman thought it was nice, but didn't want to trump the captain's yacht launch sequence from the upcoming film "Insurrection". As you remember that was a VERY dramatic, epic and cool launch sequence." 
For Insurrection, the producers decided to complete the transition into the digital realm and that this feature would be the first movie completely without the time honored motion control model photography. However, in this stage of the technique, that meant that the CGI workload had to be divided between two VFX houses, Santa Barbara Studios for the outer space shots, and Blue Sky/VIFX for the planet bound effects. As such, the model and effects, of the Cousteau, were executed for this film by Santa Barbara Studios in the Maya CGI software. (Sci-Fi & Fantasy Models, issue 34, p. 29) A relatively early outing at the time into the realm of CGI, solicited the following remark of Eaves, "The CGI guys needed every possible line drawing. They wanted all the details in advance – which differed from the way I usually work with traditional modelmakers, leaving areas open for them to contribute. But Santa Barbara did not want to have to go through an approval process on every aspect of these ships – which made sense, because there wasn't time to make that happen while still getting the ships modeled and animated and rendered out. It was a learning experience for me to provide these detailed technical drawings." (Cinefex, issue 77, p. 72) Detailed scale comparison charts, originating from this desire has made Eaves set the in-universe dimensions of the Cousteau at 110×86 feet. While perhaps not so intended, due to its predetermined position on the mother ship, the Cousteau turned out to be one of the largest small-scaled Federation vessels.
In 2015 Star Trek: The Official Starships Collection project manager Ben Robinson has indicated that the original model computer files were no longer in existence, due to Paramount Pictures's failure to maintain ownership over these, as well as due to Santa Barbara Studios' own demise, "Never found the CG models from Insurrection... so we're rebuilding them! Makes me feel like Dan Curry.", he explained, "I'm working on this [note: Son'a ships for the Collection] at the moment. It's tricky because the company that made the CG - Santa Barbara Studios didn't archive them and used their own software. Nothing is impossible though and I do have perfect reference [remark: referring to renders he had received for the Star Trek Fact Files years earlier]."  Since Robinson planned to have the Cousteau included in the Collection as one of the first two Insurrection outings, a new CGI model had to be painstakingly recreated, for which he commissioned an outside contractor, "The rebuilding work is being done by [Meshweaver's] Fabio Passaro. Luckily we have approved artwork to work from from the Fact Files days." 
True to his word, the issue for the Cousteau was released on 22 June 2016, becoming indeed the first of the Insurrection ships thus released. 
Further reading Edit
- Star Trek: The Official Starships Collection, issue 75, 2016