Did the extension to your home increase its value?
If so, the relative might have a claim in quasi-contract or unjust enrichment.
The statute of limitations may not run from the date of the construction but rather from the date that the relative demanded payment as was denied.
If this answer was helpful, please mark it as helpful or as a best answer. This answer is for general education purposes only. It neither creates an attorney-client relationship nor provides legal guidance or advice. The answer is based on the limited information provided and the answer might be different had additional information been provided. You should consult an attorney.
I assume that there was never a written agreement and that he is leaving of his own volition. If he is continuing to stay, the improvement continues to be for his benefit, not yours. While there may be other facts which would help his claim, do not assume that you are obligated to pay for the improvement. If he has lived there rent free for ten years, he may have already broken even on the deal. If, after evaluating everything, if you wish to be fair, you can work out a settlement. If so, put it in writing. Having a lawyer draft the agreement would even be better.
The foregoing answer does not constitute legal advice and does not create a lawyer-client relationship. Answers are based upon the facts stated and may change if there are additional facts not contained in the question.