hope for, wish, Hebei province
冀 centers on a deep, earnest hope or wish for something. The Hebei province meaning is a proper-noun application of that same idea of aspiration.
冀 combines 北 (originally depicting two people back-to-back, suggesting 'to turn away' or 'to hope for') with 異 (different, strange). The exact historical development is uncertain, but the character likely expressed a wish or aspiration from early times.
The top 北 looks like two people back-to-back, and the bottom 異 suggests something different or unusual. Imagine two people turning away from the ordinary and looking up with a shared wish for something better.
For キ, picture a key that unlocks a hopeful future: key -> キ, and the two people in 冀 are wishing for that key.