The traditional rule: variation would require fresh consideration
The common law rule has traditionally been that partial performance of a pre-existing debt is not good consideration1 that could bind the creditor to any new promise to excuse, discharge, or vary the original obligation of the debtor. See generally Foakes v. Beer (1884) 9 App Cas 605; Cumber v. Wane (1719) 1 Str. 426; Pinnel's Case (1602) 5 Co. Rep 117a.
However, a change in the mode of payment may be good consideration to bind the creditor to a promise to forebear from action against the debt. In Pinnel's Case, the judge said that changing the mode of payment to a "horse, hawk, or robe, &c in satisfaction is good," as would an agreement to pay early, or an agreement to pay in a different place. See also Foot v. Rawlings,  S.C.R. 197: the debtor switching to post-dated cheques was good consideration that bound the creditor.
Statutes can do away with this rule
Some jurisdictions have done away with the rule from Cumber v. Wane by statute. See e.g. British Columbia's Law and Equity Act, s. 43:
Part performance of an obligation either before or after a breach of it, when expressly accepted by the creditor in satisfaction or rendered under an agreement for that purpose, though without any new consideration, must be held to extinguish the obligation.
Courts are also reforming the doctrine of consideration
There is increasing recognition that contractual modifications may become enforceable absent fresh consideration. For example, see the estoppel-based reasoning in Central London Property Trust Ltd. v. High Trees House Ltd. (1946)  KB 130:
There has been a series of decisions over the last fifty years which, although they are said to be cases of estoppel are not really such. They are cases in which a promise was made which was intended to create legal relations and which, to the knowledge of the person making the promise, was going to be acted on by the person to whom it was made and which was in fact so acted on. In such cases the courts have said that the promise must be honoured.
At this time of day however, when law and equity have been joined together for over seventy years, principles must be reconsidered in the light of their combined effect. It is to be noticed that in the Sixth Interim Report of the Law Revision Committee, pars. 35, 40, it is recommended that such a promise as that to which I have referred, should be enforceable in law even though no consideration for it has been given by the promisee. It seems to me that, to the extent I have mentioned that result has now been achieved by the decisions of the courts.
And see Rosas v. Toca, 2018 BCCA 191 for a development that does not rely on estoppel:
The time has come to reform the doctrine of consideration as it applies in this context, and modify the pre‑existing duty rule, as so many commentators and several courts have suggested. When parties to a contract agree to vary its terms, the variation should be enforceable without fresh consideration, absent duress, unconscionability, or other public policy concerns, which would render an otherwise valid term unenforceable. A variation supported by valid consideration may continue to be enforceable for that reason, but a lack of fresh consideration will no longer be determinative. In this way the legitimate expectations of the parties can be protected. To do otherwise would be to let the doctrine of consideration work an injustice.
1. Another Q&A explains what consideration is: "It is enough that something is promised, done, forborne or suffered by the party to whom the promise is made as consideration for the promise made to him."