These were probably Convolvulus mauritanicus which is a perennial, but would need to be kept frost free over winter in a greenhouse etc. In Spring, trim off any dead bits and replant in fresh compost.

Depending where you live, they may well survive outside - I have one on a southfacing balcony that's ten years old now BUT, its not in a hanging basket, its in a tall tub, which probably gives it more protection - more soil, surrounded by plastic (tub).

Might be Ipomoea (morning glory) which is a common climber (or trailer!) in tubs/baskets etc in the summer and is treated as an annual. Its a member of the convolvulus family. Have a look for Ipomoea on a seed site (ie T+M, Nickys) there is a pale blue var.

Mine's definitely convulvulus - pic in my photos for comparison, if you want