I have a 2005 Dodge neon. It currently has 242,000 miles on the original motor. I know I'm supposed to use 5W20, but I'm wondering due to the recent heat waves with 5W-30 be acceptable
Stick with the oil weight the owner's manual specifies. Modern engines (almost anything fuel injected) aren't nearly as sensitive to outside temperatures as old engines were. My 1979 Catalina has a whole section in the owner's manual on what weight oil to use in which conditions, for example. My '99 Ranger requires one weight, nothing else. Check your owner's manual to be sure, but I doubt it specifies anything beyond a single weight.
the number after the 'w' is the viscosity of a warmed up oil. Your engine temperature is regulated , which means that viscosity will always be the same, regardless of season.
The number BEFORE the 'w' is the viscosity on cold start up. THAT's the one people change for seasons.
Do NOT change the viscosity unless you're burning or leaking a lot of oil. Some engine rely on thin oil to power sensitive hydraulic systems like valve time phasing. If your car runs normally don't mess with the recommendation or you'll just get problems.
In the winter, I use 5w30. In the summer, I use 10w40.