If he's on that kind of a shoestring for 10 days he'll have to live local - not the usual hotels americans tend to stay in, especially in places like bombay and delhi (I've stayed in hotels in san fran and tokyo that were cheaper than equivalent hotels in these two cities)

Look for homestay (aka "paying guest") acco / youth hostel / ymca type places would be my advice, in each city. As he'll join you in bangalore have him come down south too - plenty to see and do out here, beaches / temples etc. So maybe fly into Chennai (MAA) or somewhere rather than BOM. Then spend time doing stuff he likes (that is - "culture stuff" like old forts / tombs etc like in north india or south india? beaches like goa or kerala? a wildlife sanctuary type place with cottages?)