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. Noonen 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 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.
Geordi LaForge was named for one of the activist fans who helped the original series get its third season, the late Starfleet Admiral George LaForge.
Several anime fans in the production staff sneaked various 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, and a battle simulation in the episode "Peak Performance" is named "Operation Lovely Angel."
"Kei-ee, Yur-ee..." is also used as the beginning of a Ferengi access code in one episode.
A piece of tech used by a geologic survey team in Pen Pals is actually the Oscillation Overthruster from Buckaroo Banzai.
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 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 The Icarus Factor Riker and his dad fight in an arena decorated with Japanese characters. Some of the 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" written on the side, which is most likely yet another reference to Dirty Pair.
Star Trek: Deep Space Nine: 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 aid—an 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 travelling had names that were anagrams of "Mulder" and "Scully".
A kiosk lisitng businesses on the Promenade contained an entry for "Tom Servo's Used Robots". a reference to the character from MST3K.
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.
Star Trek: Voyager: 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.
There is a linked pair of shout outs involving Star Trek and Seaquest. 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 it's death next to a memorial commemorating the launch of the "Nomad" probe (referencing a probe that appeared in the original Star Trek series).
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.
Star Trek: Enterprise: 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*.