Old English belyfan "to believe,"
earlier geleafa (Mercian), gelefa (Northumbrian), gelyfan (West Saxon) "believe,"
from Proto-Germanic *ga-laubjan "to believe," perhaps literally "hold dear, love"
(cf. Old Saxon gilobian "believe,"
Old High German gilouben,
ultimately a compound based on PIE *leubh- "to care, desire, love" (see belief).
Spelling beleeve is common till 17c.; then altered, perhaps by influence of relieve, etc.
To believe on instead of in was more common in 16c. but now is a peculiarity of theology;
believe of also sometimes was used in 17c.
Related: Believed (formerly occasionally beleft);
Expression believe it or not attested by 1874;
Robert Ripley's newspaper cartoon of the same name is from 1918.
Emphatic you better believe attested from 1854.