A word of caution, however. A long time back, I used to try to do the same: try to derive obviously Hindu names always from a Sanskrit word. Thus, in Bengal, we have names like Rita, Riya, etc., which can be conceived of as distorted pronunciations of Sanskrit words (e.g. riktA, hrIYA, etc.) The only problem with that explanation is that the pattern of usage did not match: riYA seems to not be common in the subculture that would like name their daughter as `modesty', and the oldest rare riYA's I could find neither spell the name any closer to hrIYA in Bengali, nor do they know the origin of the name, as would be expected if some one picked up hrIYA in modern times. On the contrary, they seemed to belong to the subculture that could be exposed to names of middle eastern origin, and liked uncommon sounding (rather than meaning) names. Similarly, Rita seemed much more common in the Angloindian Bengali community. So, nowadays, along with looking for a Sanskrit explanation, I also look at the distribution of the names, what period and subculture it belongs to, and depending on that think of borrowing as an explanation.
Since Sajith (with the -th) is very likely South Indian or South-east Asian (or other expatriate communities), and since the name is quite common in Karnataka and Andhra Pradesh, it would help to know if the word sajjit is pronounced sajit in those communities (in the Hindi-Marathi-Bengali-Assamese-Oriya region, this would be rare: old transformations through Prakrit tend to maintain syllable weight so sajjit tends to become sAjit, which is not what I think I have heard the name pronouned as, I think it rhymes with ajit. In modern times, the -jj- is a completely common phonemic structure and not likely to undergo lenition to the different -j- in most of these languages. The other Indian languages have different, and unknown to me, phonotactis.) The antiquity of the name in these regions would also help.
On a different note, I appreciated your past contribution to this board. There are very few out here who know about names of Sanskrit origin, and a lot of misinformation propagates, so it was nice to see good information for a change.