It could be the HDD but something like this is more likely to be some age related issue. Damaged traces (the finest PCB pathways) caused by capacitor leaks. The clock capacitor in particular is renowned for leaks causing trace damage problems.
Could just be overheating; as the Xbox heats up everything expands slightly so a poor connection issue can become a no connection issue.
Identify where the trouble is can be VERY difficult but if the inside of the Xbox is carefully cleaned of dust and so on any capacitor leaks will be more easily identified.
If you're using an UnleashX dash check that the temperature threshold setting hasn't been set too low.
I doubt it is anything to do with the softmod itself but it is recommended you update that to something more recent than you probably are using at present.
