If your oversampling frequency is high (i.e. at least 1 order of magnitude higher than the sampled signal) and decimation rate low (i.e. low number of added resolution bits), you can use a simple adder to solve the problem. Example: 8bit ADC, want 10bit ADC (2 bits added resolution) -> you need to add each 2^2=4 samples and use the result as your 10bit output. For 12bit output, you would have to add batches of 2^4=16 samples, and so on.
For lower oversampling frequencies you will have to use a weighted adder instead (do a search for 'exponential average' if you don't know what is that).
Getting your hands dirty with DSP stuff is fun :)