I like to stay in the 5th or 6th. I have also stayed in the 3rd, 4th, and 7th but not as crazy about them as I am the 5th and 6th. Yes I know those are all 'touristy' but they also have a more medieval street plan and I find it certainly is central if you are going to be doing touristy things. If you are willing to look in the outer areas - 11th, 12th, etc. you will find lower prices. That said I found a number of 'decent' places under €150. Hotel St Jacques is my favorite but the price is sneaking up, not sure what it is in May. Minerve is another good one. Twice recently I've stayed at the St Pierre and love the location but the rooms are kind of small and dark (although perfectly serviceable). Hotel Marigan is another I've stayed at. That one can be super inexpensive if you don't mind sharing a toilet with one or two other rooms (shower and sink are usually in the room itself). And they do have rooms totally ensuite but then the price goes up. They also have no lift. Oh another one I like is St Andres des Artes but it's under renovation at the moment, don't think it will be done by May.
Now after looking at all those places that I've stayed in the past I just booked Hotel Abbatial St. Germain. It's a little more expensive but it has a mini-bar/fridge which I really need and most hotels in Paris do not have that.
All of these are on booking.com