There is a large debate on the origin of runes. Simply because we do not know 'exactly when, where, why or by whom runes were first created' (Findell 2014:6).
Barnes (2012: 9-10) lays out four basic points that runologists agree upon concerning the origin of runes:
The oldest runic inscriptions are dated from AD c. 150-200 and one artefact, the Meldorf fibula from northern Germany, may have a runic inscription, but may also be roman is dated to around AD c. 50.
'The centre of earliest documented runic activity is the area that comprises modern Denmark, southern Norway and southern Sweden'. A small number have been found in northern Germany and eastern Europe. In addition, all of these finds have been on portable objects. This is important because it means that these objects are found far from the places that they have been discovered.
Runes are part of the Mediterranean (phoenician-greek-etruscan-roman) alphabetic writing tradition. Runes are based on the same principles as these alphabets and 'number of runes are identical or virtually identical to characters in one or other, or several, of these alphabets (especially the roman and those of northern Italy, derived from Etruscan). They nevertheless shoe more than a little independence of the Mediterranean tradition'. For instance, the roman alphabet goes abc, runic alphabets go fuþark.
Runes were certainly derived in an area of contact with but not part of the Roman empire.
Obviously, these points are not answers, but they provide guidance when evaluating proposals on the origin of runes.
As pointed out by Barnes (2012) the majority of runologists believe that runic and various Mediterranean alphabets cannot be due to chance (10). Cultures intermixed widely during this time period and it is reasonable to assume that Germanic script derives from Mediterranean script as Germanic speakers and those who spoke with southern European tongues interacted. The arguments emerge around which alphabet runes stem from and views have changed over each decade based on new evidence. I wish I could give you a more exciting answer than Germanic peoples thought writing was a good skill and brought it back home after their travels, but this is likely the case.
I can attempt to explain some of the arguments if anyone is interested, but I am not a linguist or philologist, which are the field runology aligns more with.
Barnes, M., P. (2012) Runes: A Handbook. Woodbridge: The Boydell Press.
Findell, M. (2014). Runes. London: The British Museum.
Also, this is my first post. If I did anything wrong please let me know and I will correct it.