The Jefferies Tubes on Star Trek: The Original Series are named after Matt Jefferies, art director of the original series and designer of the original Starship Enterprise (both interiors and exterior).
And both Khan Noonien Singh and Dr. Noonien Soong (of TOS and TNG respectively) were named for a childhood friend of Gene Roddenberry. This no doubt comes as a relief to know if you were trying to figure out why their names were so similar.
The names of Clare Raymond's descendants in the Season 1 episode "The Neutral Zone" are William Hartnell, Patrick Troughton, Jon Pertwee, Tom Baker, Peter Davidson and Colin Baker. Save for one misspelling, these are the names of the first 6 actors to play The Doctor.
The Borg are one to The Cybermen. The title of their debut lampshades this by titling it "Q Who".
In one episode they mentioned a sonic screwdriver.
Several anime fans in the production staff sneaked references in:
One of the most blatant were the "exocomp" robots from "The Quality of Life". Word of God admits that their design was inspired by the Dirty Pair's Robot Buddy, Nammo. Kei and Yuri from the same series are named on various props.
"Kei-ee, Yur-ee..." is used as the beginning of a Ferengi access code in one episode.
The Enterprise has a sister ship called the Yamato, but this is actually a coincidence; they are both named after the same ship.
In Season 2's "The Icarus Factor", Riker and his dad fight in an Anbo-jyutsu arena◊ decorated with Japanese characters. Some writing is stuff you'd expect — "water", "air", "fire", etc., but a wall hanging reads "Urusai [sic] Yatsura", and you can see names of the main characters, Lum and Ataru, written on the floor if you watch carefully. "Yuri" is written on the side, another reference to Dirty Pair.
In "The Most Toys", trader Kivas Fajo is from Tomobiki City, and attended the University of Oneamisu.
In "The Price", Devinoni Ral studied at Oneamisu Campus.
There is a linked pair of shout outs involving Star Trek and Seaquest DSV. In one Next Gen episode, an on screen computer readout lists the service record of a crewman to have included the "USS Seaquest". In one Seaquest episode, an alien falls to its death next to a memorial commemorating the launch of the "Nomad" probe (referencing a probe that appeared in the original Star Trek series).
The Magnificent Ferengi was a homage to The Magnificent Seven. Also, in one episode the characters recite the first two lines of the Edwin Starr song War.
In the pilot, Bashir borrows Odo's hand for some impromptu first aidan allusion to the film M* A* S* H, in which the same thing happens to Rene Auberjonois' character.
There is also of course the well known Shout-Out in the episode "Trials and Tribble-ations" where the two officers that Temporal Investigations sent to investigate Sisko et al.'s time traveling had names that were anagrams of "Mulder" and "Scully".
A kiosk listing businesses on the Promenade contained an entry for "Tom Servo's Used Robots". a reference to the character from Mystery Science Theater 3000.
The character "Morn" is largely a background character in Quark's bar; his name is an anagram of "Norm," in homage to another familiar barfly, from Cheers.
Lounge singer Vic Fontaine is a tribute to Frank Sinatra right down to having an episode that is basically a tribute to Ocean's Eleven. And What Could Have Been: They even had Frank Sinatra, Jr. in mind for the role (but he didn't want the partnote he wanted to be on the show, but only if he could play an alien).
As revealed in "The House of Quark", Quark's father's name is Kheldar. Kheldar is also the real name of Silk, the main Drasnian character in The Belgariad. The Drasnians, like the Ferengi, have a Hat of being devious merchants.
The Okudagram graphics (created by Mike Okuda) seen in the modern Star Trek series and films are loaded with references to people and other science fiction shows and characters. Another source are the ship dedication plaques located on the bridges of Starfleet next to the turbolift. The admirals, engineers and so on listed on them are the production staff for that series. The "Most Wanted" list in Odo's office also featured photos of production staff.
In Paradise Lost when listing off the officers Sisko knows who are assigned to security work on Earth, all of them have the names of major characters from Catch22.
The Adventures of Captain Proton! Dr Chaotica wears identical clothing to Ming the Merciless, Proton's rocket ship has clear Flash Gordon influences, Satan's Robot is the "Republic Robot", an overused prop in various Republic Serials including ''Mysterious Doctor Satan'', while Proton's leather jacket with jetpack controls are the same as those used by Commando Cody.
In the episode "The Omega Directive", Seven of Nine sets up a system for capturing the Omega particle, and refers to the crew by numbers. The person who is most unhappy with this arrangement is the one assigned the number Six of Ten.
The episode "Flesh and Blood" had a brief reference to an alien race called the Ovions, who were described as hexapods. Ovions were an insect race in the Battlestar Galactica episode "Saga of a Star World".
In the teaser of "Zero Hour" the Xindi Reptilians eat live mice in a clear homage to the miniseries V.
In the series premiere, "Broken Bow", Admirals Forrest, Leonard and Williams are shout-outs to DeForest Kelley, Leonard Nimoy and William Shatner, respectively, who, of course, played McCoy, Spock and Kirk in the original series. Admiral Forrest, named after Kelley, had the most screen time throughout the series, most likely because Kelley had died in 1999, just a year or so before production began.
In the season 1 episode "Oasis," Trip sarcastically stretches the like "What do you do then? Program a holographic doctor?" *cough* Star Trek: Voyager *cough*.