It's possible the mini box is taking a long time to start up due to your Standby Power Consumption settings. Please can you check the following.
Go to Settings > System > Standby power consumption. You have 3 choices.
Fast start - the 360 starts quickly but is less energy efficient Active start - more energy efficient than fast start but the box takes a little longer to start up Eco start - the most energy efficient and probably the one you're using. It uses minimal power but takes a while to start.
The first two options are best if you have a mini box and want to watch a recording on it
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My 360 mini box doesn't switch on properly half the time. I press on, on the remote and the standby light turns white but the box doesn't always activate (The TV screen remains blank)
If I switch the power off then back on its okay.
Any ideas as what this issue is?
Are you sure the TV is turning on, don't know what lights you have on the TV, if one power is turned on and the remote is correctly paired with the TV you can quckly check if it's the TV thats not turned on by pressing the TV power button on the left of the remote.
Depending on how things have set up the TV and remote may need direct line of sight for IR operation.