Bismika Allahumma means "O Allah, with Your Name"
Here's some examples with some supplications mentioned:
Allahumma bismika amutu wa ahya
O Allah, with Your name I die and live
Whenever the Prophet (S.a.W) lay down for sleep at night, he would place his (right) hand under his (right) cheek and supplicate: "Bismika Allahumma amutu wa ahya [O ...