The orbiters were not allowed to fly through precipitation on landing for the following reasons:
The orbiter is not to encounter precipitation on any approach due to
decreased visibility, damage to the TPS, and the potential for
triggered lightning. Undesirable aspects of thunderstorms include rain
(TPS, structure), hail (TPS, structure, control), severe wind shear
(structure), turbulence (control, performance, structure), and natural
or triggered lightning (structure, electronic/software systems).
Environmental design requirements for the orbiter are based on no
in-flight penetration of thunderstorms (ref. Appendix 10-10, Volume X,
Space Shuttle Level II Program Specification). The participants in the
Weather Rules Workshop held at JSC/MSFC in October 1987 developed the
thunderstorm, lightning, and precipitation limits based on scientific
knowledge, engineering judgment, and experience.
Space Shuttle Flight Rules, precipitation rule rationale, p. 2-26
The landing weather flight rules are well summarized here:
The weather criteria are as follows:
– Cloud coverage of 4/8 or less below 8,000 feet and a visibility of 5
miles or greater required.
– The peak crosswind cannot exceed 15 knots, 12 knots at night. If
the mission duration is greater than 20 days, the limit is 12 knots,
day and night.
– Headwind cannot exceed 25 knots.
– Tailwind cannot exceed 10 knots average, 15 knots peak.
thunderstorm, lightning or precipitation activity is within 30
nautical miles of the landing site.
– At least two approach paths must
be free from detached non-transparent thunderstorm anvils less than
three-hours old within 30 nautical miles of the runway.
Turbulence must be less than or equal to moderate intensity.
– Consideration may be given for landing with a “no-go” observation
and a “go” forecast if, at decision time, analysis clearly indicates a
continuing trend of improving weather conditions, and the forecast
states that all weather criteria will be met at landing time.
The intent was clearly to wave off the deorbit burn if the criteria would not be met at landing. If the forecast for Florida landings was bad for several days, a landing could be targeted for California. (Florida landings were highly desirable to avoid the cost of ferrying the orbiter across the country.)
There is at least one documented case where mistakes were made and the orbiter landed through flight-rule-exceeding clouds (maybe technically in-limits...). That was on STS-53. The story is well told at Wayne Hale's blog here and it is worth reading for a discussion of the decision process.
Bret Copeland mentioned in comments that a similar event happened on a mission late in the program, STS-133. The commander was quoted about it:
"We have flight rules that govern the weather for landing the space
shuttle and one of the rules is the layer of clouds can't be less than
8,000 feet above the ground for landing. Unfortunately for our
landing, we broke out at 3,500 feet above the ground. So we didn't see
the runway until then," Lindsey recalled after the mission.
I am not aware of any case where the orbiter landed through precipitation.
Addressing this comment " I found few articles describing reentry/landing delays due to unfavourable weather" - there were many! Check the Space Shuttle Missions Summary and search for "flight duration changes/landing".