Pairing starts with a body built from two mirrored sides
Many animals are laid out along a left-right axis, so structures that form on both sides tend to come in twos. In vertebrates, the paired fins and limbs that let animals swim, walk, and explore new habitats are the classic example, and they develop from a specific layer of embryonic tissue on the body's flanks [2][5]. Work on lampreys, a jawless fish, found that this tissue layer, the somatic mesoderm, was already present before paired appendages appeared, suggesting the raw material for pairing existed first and appendages were built from it later [5].
The genetic side is older still. A class of developmental genes called Paired-class homeobox genes, which help set up body patterns early in animal development, has deep evolutionary roots [4]. That does not mean a single 'pairing gene' makes two of everything; it means the machinery for building patterned, two-sided bodies is ancient, and organs can be duplicated or singled out within that framework.
Twisting and coiling can turn a pair into a single organ
Pairing can also be lost. In snails and their relatives, the body twists during development (a process called torsion) and the shell coils asymmetrically, and this rearrangement is tied to the reduction or loss of organs that were originally paired, including the gills and other structures in the mantle cavity [1]. In other words, the same animal can start with two of something and end up effectively with one, not because one is useless but because the body's geometry crowded it out.
This is the clearest evidence in the supplied research for why 'two of some, one of others' is not a fixed rule. The number of an organ is a consequence of the whole body plan, and when that plan changes, organ counts change with it.
Some paired organs exist for a very specific job
Not every pair is a big, obvious organ. Insects have small accessory pulsatile organs, essentially auxiliary pumps that help circulate fluid to parts of the body the main heart cannot reach well, and these evolved multiple times with strikingly similar designs [3]. Their pairing fits the same logic: they are local solutions positioned where they are needed, on either side of the body.
The broader point is that pairing is a layout choice, not a sign of importance. A pair of tiny pumps and a pair of limbs both reflect where the body needs function, and single organs arise where one centralized structure does the job better.
How paired limbs first appeared is still an open question
The origin of paired appendages has been debated for over 150 years, and the leading ideas, including the lateral fold theory and the gill arch theory, have not been settled by decisive evidence [2]. The lamprey work adds support for the idea that the tissue needed for paired appendages existed before the appendages themselves, but it does not explain why some organs stayed single [5].
So the honest answer has two parts: the broad pattern, pairing from a two-sided body plan and losing pairs when the body twists or specializes, is well supported; the specific step-by-step origin of each paired structure is still being worked out.
Sources used in this answer
The pallial organs in the aspidobranch Gastropoda and their evolution throughout the Mollusca
Asymmetrical shell coiling and torsion in gastropods are associated with the reduction or loss of originally paired pallial organs and ctenidia.
Paired fins in vertebrate evolution and ontogeny
Paired fins and limbs are a key vertebrate adaptation, but the leading theories for their origin remain undecided after more than 150 years of debate.
Accessory pulsatile organs: evolutionary innovations in insects
Insects evolved small paired accessory pulsatile organs, auxiliary circulatory pumps, multiple times with nearly identical designs.
Evolution of homeobox genes: Q50 Paired-like genes founded the Paired class
Paired-class homeobox genes have deep evolutionary roots and play early developmental roles in patterning animal bodies.
Organization of the body wall in lampreys informs the evolution of the vertebrate paired appendages
Lamprey embryos have somatic mesoderm, the tissue that gives rise to paired appendage skeletons, indicating this tissue evolved before paired appendages appeared.