This is quite common. I was actually speaking to a leading diabetologist last about this at a meeting.
Essentially, every almost type 1 will be 'insulin resistant' in the morning. Current research indicates that this because of something called the 'dawn phenomenon', but the evidence is not conclusive.
In any case, practically what you have to do is take some extra insulin in the morning.
You have to do this even if you don't eat. This is because you are 'insulin resistant' to your background/basal insulin as well as any rapid acting insulin. However, if you don't eat a meal, you will probably become hypo. Therefore, current recommendations for type 1 diabetics (except those on pumps) is to always eat breakfast and take some extra insulin.
To remedy this situation you can either increase you insulin:carb ratio, or you can take a 'boost', or both.
As Luckystar mentions, this is all trial and error. I always recommend you make one change only at a time.
For instance, for me all day I take 1 unit of NovoRapid for 10g carbs. But at breakfast I take 2 units of NovoRapid for 10g carbs, and I also take an extra 3 units of NovoRapid regardless of the amount of carbs I have eaten.
Hope this helps.
Best, NG