What is the biblical basis for the assignment of patron saints? Within Roman Catholicism and Eastern Orthodoxy specifically (there may be more) there are many patron saints. Is there any biblical basis that exists for the assignment of a patron saint?
The word "saint" can historically and philologically be shown to originate in the Greek texts that would later be bound together as the NT, although Christianity retroactively applied the title to non-humans in the OT (like angels, as in St Michael). The invocation of saints is a phenomenon that started in the Roman period, so clearly something changed regarding the interpretation of the Greek word in Latin. 

Let's look for example at Acts 9:13:



In Ancient Greek, the word for saints here is ἁγίοις (note that throughout the NT the word is always used in, or implies, plural). The word ἅγιος appears 229 times in the Greek New Testament, and has few usages in older Greek texts. Its primary meaning is "devoted to the gods".

Note that Acts refers to a time when Christianity did not exist or was not widespread (the text is dated to around 70 CE), so the "saints" referred to here are generally held by philologists to be all Jews or Christians, i.e. God's chosen people. So in the original Greek, the word does not refer to any specific select group of Christians or Jews who served "better". This is confirmed in the translations of Acts 26:10, Romans 1:7, 1 Corinthians 1:2, and 1 Peter 1:15-16

The word ἅγιος stems from Proto-Indo-European *Hyeh₂ǵ-yus. Cognates include Latin iēiūnus (mostly used in Latin as "to sacrifice (to the Gods)", which interestingly is not used in any Latin translation although the meaning is closer to the Greek than sanctus, which is used in most Latin translations.

The word sanctus has widespread use in texts predating Christianity in Latin, mostly in Roman religion. The word divus or devus, also referring to Roman religion and frequently used as a title for Roman emperors, was used as well as a translation for ἅγιος in early texts. This is illustrated in a pamphlet, published in 1865 in Italian, titled "Sulla deificazione nel senso pagano e nel senso cattolico ... relativi all'uso della voce latina "divus" nell'epigrafia cristiana". This can be translated as "On the deification of the pagan meaning and the catholic meaning ... relative to the use of the Latin term "divus" in christian epigraphy."

So in stead of iēiūnus the Latin texts used sanctus and devus, probably because these words had more resonance with the Roman population of the day, but this is the probable root of the re-interpretation of ἅγιος implying all Jews or Christians to "a select few", and the worship of those "select few", as had been the tradition in Roman culture. Thus the worship of Saints very likely finds its roots in the deification and worship of important Roman figures like the Roman emperor, not in the original Greek NT.