I am trying to switch out speakers in my clocks, the speaker I want to get rid of is 8 ohm s and 0.5 watts, and the speaker I want to put in place of that is 8 ohms and 1 watt. Could I do this without causing any problems, or will it not have enough power?

It should work without any issues. The power rating of a speaker tells you how much power it can handle without distortion. Putting a higher-rated power speaker in place of the original one is OK.