The question of 1 year vs 2 years is the wrong question. There are advantages to both it's a decision you should make long before you are discussing with a tenant. The issue here is fear, and it's a real issue that nobody has brought up yet so I'll be the one to bring it up. If you have taken the time to figure out what kind of tenant you want, meaning credit score, income level, renal history, lease terms, ect. Then you need to hold yourself to those standards and don't make compromises. When someone asks you to change your lease terms, or rental amount you have to ask yourself why would you change your parameters? You are a smart person, a savvy investor, and have no knowledge of this individuals true motives. You did a ton of research before deciding to get into RE, and before you decided to self manage, why go against all of that for a tenant request?
And the only answer is fear, it might be a fear that you won't find a better tenant and you don't want to loose the one asking for changes. I promise there are better tenants that will fit your criteria. Maybe its a fear that your property will stay vacant longer to find the right tenant, I promise that compromising will cost you more than waiting for the right tenant. Or it could be another fear, we could go on and on here. But the advise I'm trying to give is to be true to yourself, believe in your research and decisions you made and stay in those parameters and you will do great.
You never know why they are asking, from what I read it looks like they are just trying to get a deal. If you like their profile, and want them to be your renter then you can give them a similar discount w/ a one or two year lease, the length of the lease doesn't change anything. When I get a request like this from a tenant my decision is about me and my family, not the tenant. I look at the amount and ask myself if I give this tenant a 100/mo discount, thats 24 months or 2400 I'm literally giving away. What would that same 2400 mean to my family, is that a vacation I'm giving up? or what would I have to sacrifice to give the discount? Only you can determine the value to you and your family.