In the case of Cyrillic Щ, it originated as just a scribal ligature of Ш and Т when writing Old Church Slavonic (where, naturally, it was pronounced /ʃtʲ/). It still does represent a cluster in some languages (e.g., Bulgarian /ʃt/, Ukrainian /ʃtʃ/), but in others has evolved into a single sound (e.g., [standard] Russian /ɕ:/).
The reason this particular letter stuck around is because it had a useful function: the cluster /ʃt'/ in Old Church Slavonic wasn't just any cluster, but it had an important morphophonemic function, being the jotated counterpart to /st/. In addition, Old Church Slavonic simply didn't have another good way of representing /ʃtʲ/, since palatalized /tʲ/ only occurred phonemically in this particular cluster; this probably encouraged more frequent use of the letter, although this is just me speculating.
I'm not sure, but I believe Greek ψ and ξ are similar, in that the clusters /ps/ and /ks/ have a special significance in the language, but I'm not very familiar with Greek.
Basically, morphophonemics can be a good motivator to have a single letter represent what's phonetically two sounds.