[2] Börje developed from Old Swedish Birghir which was pronounced with a voiced velar fricative [ɣ]: [birɣir].
The voiced velar fricative was spelled ⟨gh⟩ i Old Swedish[3] and changed to /j/ after /r/ in modern Svenska.
This explains why Börje has accent 2 today: since the synkope at the transition from Proto-Norse to Norse the name has been disyllabic, which leads to a word being pronounced with the grave accent in modern Swedish.
Hence, the vowel in the second syllable of old Swedish Birghir or Birgher was no svarabhakti vowel like the -e- in modern Swedish words such as the a-stem dager, which at one stage was monosyllabic (dagr) and therefore has accent 1.
[1] This "revived" form has accent 1, like an a-stem with a nominative suffix consisting of the svarabhakti-vowel -e- plus -r. Swedish names revived during romanticism commonly take a historically unjustified pronunciation.