If he's one then he's old enough for a toddler bed or even a baby bed mattress laying on the floor in an out of the way room. Be sure every bathroom door has protection on it though. Kids this age can tip over into standing water and drown very easily.
We put kids on cots as soon as they turned about 10 1/2 - 11 months old so when they moved to the toddler room on their birthday they were already used to being on the cot.
You can usually pick up cots at garage sales or even by calling some local child care centers and ask them if they have any for sale. We often sold off our older ones when we got new.
This is the style of cot we had in my child care center. We made sheets for them that were a rectangle and put several inches of elastic on each corner. That way the elastic could slip over the leg on each corner. If you want you can simply order a cot for kiddo and have it on hand then sell it later on or see if you can find one.
http://www.preschoolfurniturestore.com/Creative-Colors-To...
You can also buy a 2 inch folding pad from Walmart, the ones they take to kindergarten, and make a larger pillow case that will slip over it. Then if you have an accident on it you simply turn it over then wash it when kiddo wakes up.