Bismillahir Rahman-ir-Rahim means In the name of Allah the most loving and kind or the most gracious and merciful. What does Bismika Allahuma mean?
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 Allah, with Your Name will I die and live (wake up)]." Sahih al Bukhari