Remember when this part of the novel occurs: Jane has just gone through the traumatic revelation of finding out that Mr. Rochester is already married, and thus is unable to marry her true love. She therefore decides to flee, ignoring the part of her body ruled with passion which would have glady become Mr. Rochester's mistress somewhere in Europe. Therefore Jane is trying to escape Mr. Rochester and that relationship and move on in her life after a very painful episode. Thus all her activities are designed to prevent firstly Rochester finding her and tempting her once more to live with him as his mistress, and secondly to prevent others finding out how she is related to the incident, which surely would have been publicised when word was spread. For a woman at that time to have any stain on her reputation was a very damaging thing - it could literally ruin the rest of her life, and thus Jane decides to give the Rivers family an alias, even though she quickly admits that this is the case. Note this is not dishonesty from Jane - rather a reflection of her very difficult position. Note how she explains this position to the Rivers family:
"I fear discovery above all things; and whatever disclosure would lead to it, I avoid."
Jane is fleeing Thornfield and the events that culminated in her escape, and does not want its reach to taint her in her "new start."