Marriage Discussion will law be a problem .

It is always better to take your parents into confidence while proceeding for the matrimonial. This is not an ordinary issue but the two souls are coined together for better alliance in all the situations. The following points should be considered in such a situation-
1) You should examine her mental makeup with a prolonged association with her. A dating of at least a year will help you in knowing her attitudes, inclinations, patience and traits apart from her patience. These are the basic points and she should not fail in these parameters.
2) Relating to genetic issues, both of you should undergo for the medical tests as suggested by your doctor to ascertain Rh discrepancy. Even other tests are also required such VDRL, Blood Sugar Test, ECG test, Kidney and Liver function test so as to know normal functioning of the body. This is applicable to both of you.
3) Concurrence of the parental side for both of you is vital. Any violation in this regard would impact your normal relationship.
4) You have not indicated your financial status. You will have to take up the entire responsibilities of the family including the bringing up the kids after your marriage. You should have adequate means to take care of all such expenses coming on way after the alliance.

She is your mother's uncle's daughter. As per Sapinda relation, If the common ancestor from your father side is less than 5 generations or from the mother's side is less than three generations including you, then you both will be referred to as "Sapindas" of each other. . In these cases, the Hindu Marriage Act of 1955 bars marriage between two Hindus of this nature. But in your case, this is not applicable as yours is the 4th generation from your on your mother's side. So you need not worry about the law.
Your mother's uncle will be equal to your grandfather. His daughter will be treated as your aunt. In some families, such marriages are not approved. So you should discuss with your parents and her parents and then take their consent. If they approve your marriage will become very easy. Otherwise, you have to go against their will. You have to think and then decide. Before differing with your parents you should discuss with the girl also and what is her stand if their parents don't agree. Be sure that both of you will be on common grounds. Otherwise, you may have to face problems.
Your relation is not very close and I don't expect any genetic issues if your both marry. If you feel you can consult a doctor and show him the reports and get his opinion. The doctor will tell you and guide you properly.
Before doing all this you both should understand each other and should decide the whole issue. It should not be a one-sided issue.
I suggest you the following action plan.
1, Don't worry about the constitution and your marriage is allowed.
2. Sit with the girl and discuss the whole issue threadbare and try to understand each other. You both should come to a common understanding
2. Consult the doctor and then take his report. If you are not keen you can avoid this step.
3. Then you discuss with your family and let her discuss with their family and take their consent.
4. If the give their consent you can go ahead. If they don't give their consent both of you should review the whole issue once again and take a final call.
All the best to you

You say ;"She is my mother's uncle's daughter. " That is she is your mother's cousin.
As such she will be your aunt(once removed) and you the nephew. Hence for all common sense and practice this relationship will not be appreciated generally and especially if you follow the matriarchal tradition.
Hence it is imperative for you and the girl to discuss with your elders and then arrive at whether the marriage is permitted in your community and local society. If they agree and permit then you can go ahead.

According to me, she,the girl you wish to marry is somehow a relative to you.
First, you get confirmed whether their gothra and your gothra are not same. If it is same one you cannot marry her as she belongs to your gothra and therefore related equal to your sister.
Second see the age difference between you and her
Third, you get acceptance from your parents as well her parents
If the three above satisfies you need not see any other thing and proceed further.
The marriage between relations, though objected by science, cordial relationship you can expect in such marriage as it minimize the misunderstandings between you and her. The major misunderstanding between couples are due to the follow up in the house practices. If she is somewhat related whether close or distant, she knows the rituals, followups etc., of your family as almost similar to theirs. Many breaks in marriages are only because of mutual understanding. In some families the incoming bride very well understanding the practices of the husband's house and she very firmly attached to the family people of the husband's side. This itself a tonic to the husband.

The girl who is your choice for marriage is the cousin of your mother I think you should discuss this issue with your parents. Does she also have the same feeling for you? Have both of you discussed this issue?
I dont think you should take any step voluntarily without taking the permission of your parents and she should also take her parents consent in this regard.
If there is no issue of Gothra and other social constraints then it will be easier for you.
As far as the law is concerned The Hindu marriage law does not stop you.

As mentioned above, you are having wish to marry belong to distant relative. Since she is your mother's uncle's daughter so,she is your aunt. You would be called nephew to her.
You reside in Bihar and I know better to Bihari culture they will never allow to this marriage because I think your gotra would be same. Same gotra is not allowed in many family. Doctor also prohibit this kind of relationship because born baby with this relationship may be biological defective. As far as Hindu marriage Act 1955 is concerned, law will not disturb your marriage. As per my suggestion, both of you should discuss this issue with own family. You should also judge girl's feeling towards you as marriage is life long relation. If both of you go against the will of parents then problem may create in future.

As per Hindu Marriage Act 1955, marriages in close relations falling in Sapinda relationships are not permitted. Sapinda relationship is the extension of relations as far as 3rd generation in the line of ascent through the mother's side and as far as 5th generation in the line of ascent through the father's side in case of both the parties. The counting is done upwards as counting the involved parties as 1st generation. From that aspect you are out of that lineage and there is no legal problem on that account.

In the matters of marriages the social fabric and social customs play a very major role and you have to take cognisance of them. You are telling that this marriage being proposed by you is not a common thing in your family and you have to convince them. Nowadays, things are changing in the society and many parents are agreeing to the proposals of their children in the matters of marriages only thing is that the children should convince them about their firm determination and genuine relationship. Parents generally worry for the future lives of their children as after marriage the children may find that the decision was not correct and situation of separation had come in their lives. So you have to make your parents believe that what you are doing is doing with a firm conviction and there is good compatibility and love between you two. I hope with a firm mind and determination you can convince them.

There are cases where parents do not agree with the children and their choices. In such case you should be bold enough to take your own decisions and lead your life as per your wish. Generally, later, parents forgive the children and after seeing their happy lives just forget the past. Consider all these aspects and go ahead for the relationship.