You have more than two options... pooh pooh to that doc. I hate when docs do things like that.
I think of the three (implanon, mirena, and depo) mirena has the lowest amount of hormones. I'm not too sure about implanon. All of them use the same hormone, progestin, so you might have similar reactions to all of them. However, depo (to my knowledge) is a stronger dose so while you might feel crazy on depo, but not on mirena. (However, there is a small chance you would feel crazy on mirena too, you'll never know until you try it)
Implanon and Mirena both have similar risks too, that they might come out. Both are pretty rare. Are you comfortable touching yourself? Because with Mirena, you'll have to reach in and check the strings every month (every week for the first while) and that's kind've a pain. I'm not sure about implanon, I've heard there can be issues with scratching at the incision and the implant coming out. All of this is rare.
You could also try paraguard, which is a hormone free iud. Good option if your doc is concerned about your blood pressure. You won't have any hormonal side effects, though your period might be heavy for the first few months. In general it will stick to your normal cycle.
I'm guessing your doc just doesn't want you on estrogen- in that case, you could try pop's too (progesterone only pills, also known as mini-pills). The downside to mini pills is that you have to be really really careful about taking them at the same time every day.
There are other things too, such as a diaphragm used with spermicide, condoms, etc. Those wouldn't effect your blood pressure and many women rely on them.
My suggestion is to go to a website such as planned parenthood's site- they give you lots of options for different types of birth control and what the good things and bad things are for each type.
Every type of birth control affects every woman differently, and unfortunately, you'll never know until you try them. I will tell you that most women are happy with the type they choose. Mirena is really easy for your doc to remove if you're not happy... I don't think implanon is as easy to remove. Mirena lasts for 5 years, implanon for 3. It's your choice!