Solution 1:

The OED’s etymology entry notes that it is abnormal, but does not say how this came to be.

Etymology: Com. Teut. and Indogermanic: OE. siwan, siowan (usually, with change of conjugation, siwian, siowian, seowian) = OFris. sîa (mod.Fris. dial. siije), OHG. siuwen, ONor. sýja (Sw. sy, Da. sye), Goth. siujan :— OTeut. *siwjan, cogn. w. the synonymous L. su-ĕre, Gr. (κασ-)σύειν, Lett. schuju, OSl. šiti (Russ. shitp., shivatp.), Skr. siv (3 sing. pres. sīvyati, pa. pple. syūtá; derivatives are syū fem., needle or thread, syūman suture).

The root (for which Hirt suggests a primary form *seyewa-) appears in the words above quoted as *syū- : *sīw. Another ablaut-grade, *syou-, is found in OTeut. *saumo- seam sb. The pronunciation [səʊ] is abnormal (cf. strow, var. of strew, repr. OE. streowian); the written forms show that it goes back at least to the 14th c. In the 17th c. sew sometimes rhymes with clue, new; the mod.Sc. pronunciation is [ʃu].