2005 hyundai tucson blows cold air on right side when heater is on

SOURCE: Why doesn't the heater blow hot air?

If your "Check Engine" or "Service Engine Soon" light is on, you need to have that checked out first.
If not, the next thing you should check is your engine coolant level. (Antifreeze) If the coolant level is low, the engine coolant does not circulate through the heater core and the the heater doesn't work. The coolant level could be low for a number of reasons including a leaky hose, water pump, or head gasket. In any case, if it is found to be low, you need to find out why.
If the coolant level is OK then the thermostat function needs to be checked to make sure that the engine is reaching operating temperature.
If all of the above checks out OK, then the heater control valve and/or the blend air door in the HVAC housing needs to be checked for proper operation. (NOTE: some cars are not equipped with a heater control valve)

SOURCE: how many bolts on the thermostat housing?

should just be two bolts, one on each end of the housing

SOURCE: Hyundai Tucson a/c starts out

The issue you describe can be caused by several possible conditions. The best advice I can offer is take the vehicle into an authorized Hyundai Service Department to have it properly diagnosed.
The worse thing about chasing parts is you may or may not just get lucky on the repair. Then again you may end up shelling out $100's of bucks before you actually hit on the cause.
Possible Causes
1. Malfunctioning thermostat in the evaporator - monitors evaporator temperature so it doesn't freeze over.
2. Malfunctioning A/C control head (panel)
3. System is possibly UNDERcharged (insufficient refrigerant)
4. Malfunctioning Tripple Switch (the switch monitors pressure in the system: i.e., if pressure is too high, it will shut down the A/C; if pressure is too LOW, it will shut down the A/C) If this switch is malfunctioning, it can cause the symptom you describe.
5. Engine Coolant Temperature Sensor malfunction - if the sensor is lying, telling the engine computer that the temperature of the engine is above 230 degrees, the computer will shut down the A/C.
This is why you should have the problem properly diagnosed.
Hope this helps.